Home OD-characterization of alternating groups Ap+d
Article Open Access

OD-characterization of alternating groups Ap+d

  • Yong Yang EMAIL logo , Shitian Liu and Zhanghua Zhang
Published/Copyright: August 19, 2017
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

Abstract

Let An be an alternating group of degree n. Some authors have proved that A10, A147 and A189 cannot be OD-characterizable. On the other hand, others have shown that A16, A23+4, and A23+5 are OD-characterizable. We will prove that the alternating groups Ap+d except A10, are OD-characterizable, where p is a prime and d is a prime or equals to 4. This result generalizes other results.

Keywords: Order component; Element order; Alternating group; Degree pattern; Prime graph; Simple group
MSC 2010: 20D05; 20D06; 20D60

1 Introduction

All groups under consideration are finite and for a simple group we mean a non-abelian simple group. For a given group G, the socle of G is defined as the subgroups generated by the minimal normal subgroups of G, denoted by Soc(G). π(G) is the set of prime divisors of |G|. Sylp(G) is the set of all Sylow p-subgroups of G, where p ∈ π(G), Sylr(G) is the set of the Sylow r-subgroups of G for r ∈ π(G). Let Aut(G) and Out(G) denote the automorphism and outer-automorphism of G, respectively. Sn and An denote the symmetric and alternating groups of degree n, respectively. ω(G) is the set of orders of its elements of G. Associated to ω(G) a graph is named prime graph of G, which is written by GK(G). The vertex set of GK(G) is π(G), and two distinct vertices p, q are joined by an edge if p ⋅ q ∈ ω(G) which is denoted by p ∼ q. We use s(G) to denote the number of connected components of the prime graph GK(G). If n = pam with (p, m) = 1, then np = pa. The other symbols are standard (see [1], for instance).

Definition 1.1

([2]). Let G be a finite group and |G|=p1α1p2α2⋯pkαk, where pis are different primes and αis are positive integers. For p ∈ π(G), let deg(p) := |{q ∈ π(G)|p ∼ q which we call the degree of p. We also define D(G) := (deg(p1), deg(p2), ⋯, deg(pk)), where p1 < p2 < ⋯ < pk. We call D(G) the degree pattern of G.

Given a finite group M, denote by hOD(M) the number of isomorphism classes of finite groups G such that (1) |G| = |M| and (2) D(G) = D(M).

Definition 1.2

([2]). A finite group M is called k-fold OD-characterizable if hOD(M) = k. Moreover, a 1-fold OD-characterizable group is simply called an OD-characterizable group.

For the groups which are k-fold OD-characterizable, please see Tables 1 and 2 and corresponding references of [3]. Many finite groups are k-fold OD-characterizable.

Table 1

The simple classical groups

L Lie;rankL d O |L|
Ln(q) An − 1(q) (n, q − 1) 2 df, if n ≥ 3; 1dqn(n−1〉/2∏i=2n(qi−1)
n − 1 df, if n = 2
Un(q) 2An − 1(q) (n, q + 1) 2 df, if n ≥ 3 1dqn(n−1〉/2∏i=2n(qi−(−1)i)
[n/2] df, if n = 2
PSp2m(q) Cm(q) (2, q − 1) df, m ≥ 3; 1dqm2∏i=1m(q2i−1)
m 2f, if m = 2
Ω2m+1(q) Bm (q) 2 2f 12qm2∏i=1m(q2i−1)
q odd m
PΩ2m+(q) Dm (q) (4, qm − 1) 2df, if m ≠ 4 1dqm(m−1)(qm−1)Πi=1m−1(q2i−1)
m ≥ 3 m 6df, if m = 4
PΩ2m−(q) 2Dm(q) (4, qm + 1) 2 df 1dqm(m−1)(qm+1)Πi=1m−1(q2i−1)
m ≥ 2 m − 1

Table 2

The simple exceptional groups

L L d O |L|
G2(q) 2 1 f, if p ≠ 3 q6(q2 − 1)(q6 − 1)
2f, if p = 3
F4(q) 4 1 (2, p)f q24(q2 − 1)(q6 − 1)(q8 − 1)(q12 − 1)
E6(q) 6 (3, q − 1) 2df 1dq36∏i∈{2,5,6,8,9,12}(qi−1)
E7(q) 7 (2, q − 1) df 1dq63∏i∈{2,6,8,10,12,14,18}(qi−1)
E8(q) 8 1 f q120∏i∈{2,8,12,14,18,20,24,30}(qi−1)
2B2(q), q = 22m+1 1 1 f q2(q2 + 1)(q − 1)}
2G2(q), q = 32m+1 1 1 f q3(q3 + 1)(q − 1)
2F4(q), q = 22m+1 2 1 f q12(q6 + 1)(q4 − 1)(q3 + 1)(q − 1)
3D4(q) 2 1 3f q12(q8 + q4 + 1)(q6 − 1)(q2 − 1)
2E6(q) 4 (3, q + 1) 2df 1dq36∏i∈{2,5,6,8,9,12}(qi−(−1)i)

Some authors have proved that some alternating groups with s(G) ≥ 2 are OD-characterizable.

Proposition 1.3

The alternating groups Ap, Ap+1 and Ap+2, where p is a prime, are OD-characterizable [4].

But if s(G) = 1, then what is happening to the influence of OD on the structure of groups? Recently the following results have been shown.

Proposition 1.4

Let An be alternating group of degree n. Then the following statements hold.

  1. Ap+3, where p is a prime and 7 ≠ p, are OD-characterizable [5-7].

  2. A10 is 2-fold OD-characterizable [8].

  3. An with that either 5 ≤ n ≤ 100 and n ≠ 10 or n = 106, 112, is OD-characterizable [5, 7, 9-12].

  4. Ap+5 with p ≤ 1000 a prime is OD-characterizable [13].

  5. A147 and A189 are 7-fold and 14-fold OD-characterizable, respectively [14].

  6. Assume that p is a prime satisfying the following three conditions:

    (6a) p ≠ 139 and p ≠ 181,

    (6b) π((p + 8)!) = π(p!),

    (6c) p ≤ 997.

    Then Ap+8 is OD-characterizable [14].

Form Proposition 1.4(2)(3), some alternating groups cannot be OD-characterizable. So naturally, we will ask this question: which of the alternating groups is OD-characterizable? Thus in this paper, we give a new characterization of alternating group Ap+d by OD. In fact, we will prove the following result.

Main theorem 1.5

Let p be a prime and d positive. If d is equal to 2 or 4 or d is a prime, then alternating groups Ap+d except A10 are OD-characterizable.

This result confirms the conjecture [15, Conjecture 5.1] put forward by of Yan et al.

2 Some lemmas

In this section, we will give some results which will be used.

Lemma 2.1

Let An (or Sn) be an alternating (or a symmetric group) of degree n. Then the following hold.

  1. Let p, q ∈ π(An) be odd primes. Then p ∼ q if and only if p + q ≤ n.

  2. Let p ∈ π(An) be odd prime. Then 2 ∼ p if and only if p + 4 ≤ n.

  3. Let p, q ∈ π(Sn). Then p ∼ q if and only if p + q ≤ n.

Proof

It is easy to get from [16].□

By [1], we have that |An| = n!/2 and |Sn| = n!.

Let exp(n, r) = a or ra‖ n, where a is a positive integer satisfying ra|n but ra+1 ∤ n.

Lemma 2.2

Let Ap+k be an alternating group of degree p + k such that p > k ≥ 3 and p + i is composite, i = 3, ⋯, k, where p is a prime. Let |π(Ap + k)| = d. Then the following hold.

  1. deg(2) = deg(3) = d − 1 if k ≥ 4; deg(2) = d − 2 and deg(3) = d − 1 if k = 3.

  2. deg(p) = |π(k!)| if k ≥ 4; deg(p) = |π(k!)| − 1 if k = 3.

  3. exp⁡(|Ap+k|,2)=∑i=1∞[p+k2i]−1. In particular, exp(|Ap+k|, 2) ≤ p + k − 1.

  4. exp⁡(|Ap+k|,r)=∑i=1∞[p+kri] for each r ∈ π(Ap+k)\{2}. Furthermore, exp⁡(|Ap+k|,r)<p+k2, where 3 ≤ r ∈ π(Ap+k). In particular, if r>[p+k2],thenexp⁡(|Ap+k|,r)=1.

Proof

  1. (1) By Lemma 2.1, we have that 2 ∼ p and 3 ∼ p. Hence deg(2) = d − 1 = deg(3).

  2. For r ∈ π(Ap+k)\{2, p}, by Lemma 2.1, it is easy to see that r ∼ p if and only if p + r ≤ p + k. It follows that r ≤ k and r ∈ π(k!). Hence deg(p) = |π(k!)|.

  3. By the definition of Gaussian integer function, we have that

    exp⁡(|Ap+k|,2)=∑i=1∞[p+k2i]−1=([p+k2]+[p+k22]+[p+k23]+⋯)−1≤(p+k2+p+k22+p+k23+⋯)−1=(p+k)(12+122+123+⋯)−1=p+k−1.

  4. By the same as in proof of (3), we have that

    exp⁡(|Ap+k|,r)≤(p+k)(1r+1r2+1r3+⋯)=p+kr−1≤p+k2

    for an odd prime r∈π(Ap+k).Ifr>[p+k2],exp⁡(|Ap+k|,r)=1.

The proof is completed.□

Lemma 2.3

Let L be a nonabelian simple group. Then the order and its outer-automorphism of L are as listed in Tables 1−3.

Table 3

The simple sporadic groups

L d O |L|
M11 1 1 24⋅ 32⋅ 5⋅ 11
M12 2 2 26⋅ 33⋅ 5⋅ 11
M22 12 2 27⋅ 32⋅ 5⋅ 7⋅ 11
M23 1 1 27⋅ 32⋅ 5⋅ 7⋅ 11⋅ 23
M24 1 1 210⋅ 33⋅ 5.7⋅ 11⋅ 23
J1 1 1 23⋅ 3⋅ 5⋅ 7⋅ 11⋅ 19
J2 2 2 27⋅ 33⋅ 52⋅ 7
J3 3 2 27⋅ 35⋅ 5⋅ 17⋅ 19
J4 1 1 221⋅ 33⋅ 5⋅ 7⋅ 113⋅ 23⋅ 29⋅ 31⋅ 37⋅ 43
HS 2 2 29⋅ 32⋅ 53⋅ 7⋅ 11
Suz 6 2 213⋅ 37⋅ 52⋅ 7⋅ 11⋅ 13
McL 3 2 27⋅ 36⋅ 53⋅ 7⋅ 11
Ru 2 1 214⋅ 33⋅ 53⋅ 7⋅ 13⋅ 29
He (F7) 1 2 210⋅ 33⋅ 52⋅ 73⋅ 17
Ly 1 1 28⋅ 37⋅ 56⋅ 7⋅ 11⋅ 31⋅ 37⋅ 67
ON 3 2 29⋅ 34⋅ 5⋅ 73⋅ 11⋅ 19⋅ 31
Co1 2 1 221⋅ 39⋅ 54⋅ 72⋅ 11⋅ 13⋅ 23
Co2 1 1 218⋅ 36⋅ 53⋅ 7⋅ 11⋅ 23
Co3 1 1 210⋅ 37⋅ 53⋅ 7⋅ 11⋅ 23
Fi22 6 2 217⋅ 39⋅ 52⋅ 7⋅ 11⋅ 13
Fi23 1 1 218⋅ 313⋅ 52⋅ 7⋅ 11⋅ 13⋅ 17⋅ 23
Fi24′ 3 2 221⋅ 316⋅ 52⋅ 73⋅ 11⋅ 13⋅ 17⋅ 23⋅ 29
HN(F5) 1 2 214⋅ 36⋅ 56⋅ 7⋅ 11⋅ 19
Th(F3) 1 1 215⋅ 310⋅ 53⋅ 72⋅ 13⋅ 19⋅ 31
BM(F2) 2 1 241⋅ 313⋅ 56⋅ 72⋅ 11⋅ 13⋅ 17⋅ 19⋅ 23⋅ 31⋅ 47
M(F1) 1 1 246⋅ 320⋅ 59⋅ 76⋅ 112⋅ 133⋅ 17⋅ 19⋅ 23⋅ 29⋅ 31⋅ 41⋅ 47⋅ 59⋅ 71

Proof

See [17].□

Lemma 2.4

If n≥ 6 is a natural number, then there are at least s(n) prime numbers pi such that n+12<pi<n.

Here

  1. s(n) = 6 for n ≥ 48;

  2. s(n) = 5 for 42 ≤ n ≤ 47;

  3. s(n) = 4 for 38 ≤ n ≤ 41;

  4. s(n) = 3 for 18 ≤ n ≤ 37;

  5. s(n) = 2 for 14 ≤ n ≤ 17;

  6. s(n) = 1 for 6 ≤ n ≤ 13.

In particular for every natural number n > 6, there exists a prime p such that n+12<p<n−1, and for every natural number n > 3, there exists an odd prime number p such that n − p < p < n.

Proof

See Lemma 1 of [18].□

Lemma 2.5

Let G be a finite non-abelian simple group and p is the largest prime divisor of |G| with p ‖|G|. Then p||Out(G)|.

Proof

We know that if a prime p divides the order of outer-automorphism of a group G, then the Sylow p-subgroups of G are noncyclic (see [19, Corollary 11.21]).□

Lemma 2.6

Let G be a simple group of Lie type in characteristic p. Assume that pk||G| but pk+1 ∤ |G| where k is a positive integer, then |G| < p3k, and if G ≇ L2(q) with q = pk, then |G|<p83k.

Proof

It is easy to get from Tables 1 and 2 of Lemma 2.3.□

3 Proof of the main theorem

In this section, we give the proof of main theorem.

For easy reading, we rewrite the main theorem.

Theorem 3.1

Let p be a prime. Assume that d is a positive integer such that either d is a prime if d is odd or d is equal to at most 4 if d is even. Then alternating groups Ap+d except A10 are OD-characterizable.

Proof

By Proposition 1.4(1), we can assume that d ≥ 4. Propositions 1.3 and 1.4(3)(5) implies that if p + d is a prime, then the alternating group is OD-characterizable, hence we always consider the case: π((p + d)!) = π(p!) and p ≥ 113. We have divided the proof of theorem into a series of lemmas. Note that p is always the largest prime divisor of |G|.

Step 1: G is insoluble.

Proof

Since n ≥ 118 and |G|=n!2, then by Lemmas 2.2 and 2.4 there exists a prime p1 such that |G|p1 = p1, and p1 ∤ (p − 1). Assume that G is soluble, then there is a subgroup H of order p1 p. Since p1 ∤ (p − 1), then H is cyclic and so G has an element of order p1 p. It follows that deg(p) > |π(d!)|, a contradiction to Lemma 2.2. Therefore G is insoluble.□

We use the notation of [20]. Let |An|=n!2=q1α1q2α2⋯qkαkwhereα1=∑s=1∞[n2s]−1andαi=∑s=1∞[nqis], 2 ≤ i ≤ k and the qi are prime numbers. If n2<p and n ≥ 113, then by Lemma 2.4, αi = 1 for i ∈ {k − 5, k − 4, k − 3, k − 2, k − 1, k}. For convenience, we set tn(1)=∏n2<p1≤np1,tn(2)=∏n3<p2<n2p22tn(1),⋯, and tn(k)=∏i=1k(∏ni+1<pi<nipi)i where pj = 1 if there is no prime between nj+1andnj.

Step 2: There exists a normal series 1 < K < H < G such that s ≲ G/K ≲ Aut(S) where S ≅ H/K.

Let G = G/K and S = Soc(G). Then S = B1 × B2 × ⋯ × Bm, where the Bi are non-abelian simple groups and S ≲ G ≲ Aut(S). In the following, we will prove that m = 1.

Assume that m ≥ 2. Then there is a prime p | tn(1) such that p ∤ |S|. Otherwise, by Lemma 2.4 there is a prime p′ | tn(1) with that pp′ ∈ ω(G), in contradiction to Lemma 2.2 (since p′ ∤ (p − 1) and p′ ∉ π(K), then p′ ∈ π(CG/K(SK/K))). Thus for every i, Bi ∈ 𝔉p′, where p′ < p is the second maximal prime of π(G) and 𝔉p′ is the set of non-abelian finite simple group S such that p′∈π(S)⫅{2,3,5,⋯,p′}.

We shall prove that p ∤ |K|. Otherwise, by NC theorem, NG(Kp)/CG(Kp) is isomorphic to a subgroup of Aut(Kp) where Kp is a Sylow p-subgroup of K and also a Sylow p-subgroup of G. Then there is a prime p′ that |G|p′ = p′ divides |K|. It follows that there exists an element of order pp′ contradicting to deg(p) = |π(d!)| by Lemma 2.2. Hence p||Out(S)|. We know that

Out(S)=Out(S1)×Out(S2)×⋯×Out(Sr),

where the groups Sj(j = 1, 2, ⋯, r) are direct products of all isomorphic Bi′ s such that

S=S1×S2×⋯×Sr.

Therefore for certain j, p divides the order of an outer-automorphism of a direct product Sj of t isomorphic simple groups Bi for some 1 ≤ j ≤ m. Since Bi ∈ 𝔉p, it follows from Lemma 2.5, that p ∤ |Out(Bi)|. But by [21], |Aut(S)| = |Aut(Sj)|t ⋅ t!. Thus t ≥ p. Since Bj is non-abelian simple group, then 4p||Aut(Bi)|t, and hence, 22p||G| which contradicts Lemma 2.2. Hence m = 1 and S = Bi.

Let |π(n)| be the number of primes ≤ n.

Step 3: tn(1)||S|. In particular, if n ≥ 118, then tn(9)||S|.

First we show that tn(1) divides the order |H/K| of H/K. We know that |Gp| = p, and so we can assume that p divides H/K and p ∤ |K|. Then H/K is nonabelian. If not, H/K is a group of order p. By NC theorem, G/KCG/K(H/K)≅ a subgroup of Zp − 1 where Zn is a cyclic group of order n. As n ≥ 118, there exists another prime p′ (≠ p) dividing tn(1). Since p′ ∤ p − 1 and p′ ∉ π(K), p′ ∈ π(CG/K(H/K)) and so pp′∈ω(G) contradicting to deg(p) = |π(q)|. Thus H/K is non-abelian. If there is a prime p′ ∉ π(H/K) dividing tn(1), then p′ ∈ π(G/H). By Frattini’s argument, G = NG(P)H where P ∈ Sylp(H). Now we have NG(P)CG(P)≅ a subgroup of Zp − 1 and so p′ ∈ π(CG(P)), whence p p′ ∈ ω(G) a contradiction since deg(p) = |π(d!)|. Therefore, tn(1) divides the order |H/K| of H/K.

Secondly, we prove tn(9)||S|. If n is ≥ 118, then there are two primes q1, q2 such that 910n<q2<q1≤n. Indeed, using Theorem 1 of [22],

xlog⁡x(1+l2log⁡x)<|π(x)|<xlog⁡x(1+32log⁡x).

Hence we have

|π(n)|−|π(910n)|>nlog⁡n(1+l2log⁡n)−910nlog⁡910n(1+32log⁡910n).

By direct computation, |π(n)|−|π(910n)|>1 when n ≥ 18000. While 118 ≤ n < 18000, we can check it directly by [23].

Let n10<p1≤n2 where p1 is a prime. If p1||S|, then p1||K| or p1||G/H|. If p1||G/H|, then G has an element of order p1 p, a contradiction according to the above arguments. If p1||K|,thenp1j||K| for j ∈ {2, 3, 4, 5, 6, 7, 8, 9}. We conclude that there is a subgroup M of order ppi for i ≤ 9. Obviously, G has no element of order ppi as deg(p) = |π(d!)| and by Frattini’s argument. By Theorem 9.3.1 of [24], pi| (p − 1) for i ≤ 9. Then p1 p2 ⋯ p9| (p − 1). But n2≥q1>q2>⋯>q9>n10>p10andhence,∏i=19qi>p by direct check when p ≥ 113, a contradiction. Therefore tn(9) divides the order |S| of S.

Step 4: If n ≥ 118, then tn(9) > 7.82n.

If n ≥ 118, then by Theorem 1 of [22], |π(ni)|>nilog⁡ni(1+l2log⁡ni)fori∈{1,2,⋯,9} and by Corollary 2 of [22], |π(n)|<5n4log⁡n. Thus for n ≥ 15

log⁡tn(9)=∑n2<p 2≤nlog⁡p1+2∑n3<p 9≤n2log⁡p2+⋯+9∑n10<p 1≤n9logp9>(|π(n)|−|π(n2)|)log⁡n2+2(|π(n2|)−|π(n3)|)log⁡n3+⋯+9(|π(n9)|−|π(n10)|)log⁡n10=|π(n)|log⁡n2+|π(n2)|(2log⁡n3−log⁡n2)+⋯+|π(n9)|(6log⁡n7−5log⁡n6)−6|π(n7)|log⁡(n7)>nlog⁡n(1+l2log⁡n)log⁡(n2)+n2log⁡n2(1+l2log⁡n2)log⁡(2n32)+n9log⁡n9(1+l2log⁡n9)log⁡98n109−9⋅1.25⋅n10=∑i=19nilog⁡ni(1+l2log⁡ni)log⁡ii−1n(i+1)i−1.25⋅9n10=18.93167097−16.875>2.05667n

So we conclude that tn(9) > e2.05667n > (7.82)n.

Step 5: S is isomorphic to An with n = p, p + 1, p + 2, p + 3, ⋯, p + d.

According to Classification Theorem of Finite Simple Groups and Step 2, S is isomorphic to a sporadic simple group, a simple group of Lie type or an alternating group. If S is isomorphic to a sporadic simple group and p ≥ 113, then there is no group satisfying tn(9)||S| by Table 3. If S is a simple group of Lie type in characteristic p, then p≤n10. If not and we assume that n2<p≤n,thentn(1)>(|π(n)|−|π(n2)|)(n2)>(n2)8 (By corollary 3 of [22], |π(n)|−|π(n2)|>3⋅n25⋅log⁡n2>8forn≥118). So tn(1) > n7 > p7, a contradiction to Lemma 2.6(p||S| by Step 3). If nk+1<p≤nk with k = 2, 3, 4, then pk||S| by Step 3 and so |S| < p3k by Lemma 2.6. Thus |S|pk<p2k<p10<(n2)10<tn(1) and hence |S| < pktn(1) < tn(9), a contradiction since tn(9) divides the order |S| of S by Step 3. If k = 5, 6, 7, 8, 9, then similarly we get |S|pk<p2k<p18<(n5)18<tn(5) and so |S| < pktn(5) < tn(9), a contradiction (when n≥118,|π(nk)|−|π(nk+1)|>1fork∈{5,6,7,8,9}).

Let pl||S|. If p ≥ 3, then by Lemmas 2.6 and Step 4,

7.82n<tn(9)≤|S|pl<p2l≤(p∑i=1∞[npi])2=(pnp−1)2<(323−1)n=3n

(if x ≥ 3 and f(x)=x1x−1, then the differentiate f′(x) of f(x) is < 0), a contradiction since 7. 82 ≰ 3. If p = 2 and S ≇ L2(2l), then 7.82n<tn(9)<|S|2l<25l3≤(2∑i=1∞[npi])53=(253)n and so 7. 82 < 25/3 < 22 = 4, a contradiction. If S ≅ L2(2l), then S contains maximal cyclic subgroups of orders pl + 1 and pl − 1 and so S contains at most two prime factors which are larger than n/2, a contradiction to the fact that |π(n)|−|π(n2)|>8 for n ≥ 118.

Since p is the largest prime divisor of |G| and by Step 3, p divides |S|, then S is isomorphic to An for n = p, p + 1, ⋯, p + d.

Step 6: G is isomorphic to Ap+d.

The proof of Step 5 implies that S is isomorphic to An with n = p, p + 1, ⋯, p + d. By Steps 2 and 5, we have that An ≤ G/K ≤ Sn.

If H/K ≅ Ap, then Ap ≤ G/K ≤ Sp. If G/K ≅ Ap, then |K| = (p + 1)(p + 2) ⋯ (p + d). It is easy to see that there is a prime divisor p′ of |K| such that p′ > d and p′ ∤ p − 1. Note that p is the largest prime of π(G) and d ≥ 4. Then there is cyclic subgroup of order pp′ and so p′ is adjacent to p. It follows that pp′ ∈ ω(G) and so deg(p) > |π(d!)|, a contradiction. If G/K ≅ Sp, then |K| = (p + 1)(p + 2) ⋯ (p + d)/2. In this case, we also get that there exists a prime p′ of |K| with the properties: p′ > d and p′ ∤ p − 1. We similarly get that pp′ ∈ ω(G) and so we have a contradiction as deg(p) = |π(d!)|.

Similarly to the above arguments, we also can rule out these cases: H/K ≅ An with n ∈ {p + 1, p + 2, ⋯, p + d − 1}.

If H/K≅ Ap+d, then Ap+d ≤ G/K ≤ Sp+d. If G/K≅ Ap+d, then K = 1, the desired result. If G/K≅ Sp+d, then |G|2 < |Sp+d|2, a contradiction.

Steps 1-6 complete the proof of the Maintheorem 1.5.□

By Propositions 1.3 and 1.4, some alternating groups are OD-characterizable and by our main theorem 1.5 we have the following.

Theorem 3.2

Let p be a prime. Then the alternating group Ap+d except A10 where d is a prime or equals to 4, is OD-characterizable.

The conjecture in [25] was proved true by some joint works of many mathematicians and the last important part of the proof was given by V.D.Mazurov, Chen, SHI, etc. (see [26] and related references). Therefore we can get the following theorem which is also a conjecture.

Theorem 3.3

Let G be a group and H a finite simple group. Then G ≅ H if and only if (a) ω(G) = ω(H) and (b) |G| = |H|.

Corollary 3.4

Let G be a group and p ≥ 5 a prime. Assume that d is either a prime or 4. Then G ≅ Ap+d if and only if ω(G) = ω(Ap+d) and |G| = |Ap+d|.

Acknowledgement

The corresponding author was supported by the Department of Education of Sichuan Province (Grant No: 15ZA0235) and by the Opening Project of Sichuan Province University Key Laborstory of Bridge Nondestruction Detecting and Engineering Computing (Grant Nos: 2016QYJ01 and 2017QZJ01). The authors are very grateful for the helpful suggestions of the referee.

  1. Conflict of interest

    The conflict of interest disclosure: The authors declare that there is no conflict of interest regarding the publication of this paper.

References

[1] Conway J. H., Curtis R. T., Norton S. P., Parker R. A., Wilson R. A., Atlas of finite groups. Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups, With computational assistance from J. G. Thackray.Search in Google Scholar

[2] Moghaddamfar A. R., Zokayi A. R., Darafsheh M. R., A characterization of finite simple groups by the degrees of vertices of their prime graphs. Algebra Colloq., 2005, 12(3), 431–442.10.1142/S1005386705000398Search in Google Scholar

[3] Akbari M., Moghaddamfar A. R., Some quantitative characterizations of certain symplectic groups over the binary field. Int. Electron. J. Algebra 2014, 16, 32–52.10.24330/ieja.266225Search in Google Scholar

[4] Moghaddamfar A. R., Zokayi A. R., Recognizing finite groups through order and degree pattern. Algebra Colloq., 2008, 15(3), 449–456.10.1142/S1005386708000424Search in Google Scholar

[5] Hoseini A. A., Moghaddamfar A. R., Recognizing alternating groups Ap+3 for certain primes p by their orders and degree patterns. Front. Math. China, 2010, 5(3), 541–553.10.1007/s11464-010-0011-ySearch in Google Scholar

[6] Liu S., OD-characterization of some alternating groups. Turkish J. Math., 2015, 39(3), 395–407.10.3906/mat-1407-53Search in Google Scholar

[7] Moghaddamfar A. R., Rahbariyan, S., More on the OD-characterizability of a finite group. Algebra Colloq., 2011, 18(4), 663–674.10.1142/S1005386711000514Search in Google Scholar

[8] Zhang L., Shi, W., OD-characterization of simplex K4-groups. Algebra Colloq., 2009, 16(2), 275–282.10.1142/S1005386709000273Search in Google Scholar

[9] Kogani-Moghaddam R., Moghaddamfar, A., Groups with the same order and degree pattern. Sci. China Math., 2012, 55(4), 701–720.10.1007/s11425-011-4314-6Search in Google Scholar

[10] Moghaddamfar A. R., Zokayi A. R., OD-characterization of alternating and symmetric groups of degrees 16 and 22. Front. Math. China, 2009, 4(4), 669–680.10.1007/s11464-009-0037-1Search in Google Scholar

[11] Yan Y., Chen G., OD-characterization of alternating and symmetric groups of degree 106 and 112. In Proceedings of the International Conference on Algebra 2010 (2012), World Sci. Publ., Hackensack, NJ, pp. 690–696.Search in Google Scholar

[12] Zhang L. C., Shi W. J., OD-characterization of A16. J. Suzhou Univ. (Natural Sci. Edi.), 2008, 24(2), 7–10.Search in Google Scholar

[13] Yan Y., Xu H., Chen G., OD-characterization of alternating and symmetric groups of degree p + 5. Chin. Ann. Math. Ser. B, 2015, 36(6), 1001–1010.10.1007/s11401-015-0923-7Search in Google Scholar

[14] Liu S., Zhang Z., A characterization of some alternating groups Ap+8 of degree p + 8 by OD. SpringerPlus, 2016, 5(1), 1–10.10.1186/s40064-016-2763-7Search in Google Scholar PubMed PubMed Central

[15] Yan Y., Chen G., Zhang L., Xu, H., Recognizing finite groups through order and degree patterns. Chin. Ann. Math. Ser. B, 2013, 34(5), 777–790.10.1007/s11401-013-0787-7Search in Google Scholar

[16] Zavarnitsin A. V., Mazurov V. D., Element orders in coverings of symmetric and alternating groups. Algebra Log., 1999, 38(3), 296–315.10.1007/BF02671740Search in Google Scholar

[17] Kleidman, P., Liebeck M., The subgroup structure of the finite classical groups, vol. 129 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1990.10.1017/CBO9780511629235Search in Google Scholar

[18] Kondrat’ev A. S., Mazurov V. D., Recognition of alternating groups of prime degree from the orders of their elements. Sibirsk. Mat. Zh., 2000, 41(2), 359–369, iii.10.1007/BF02674599Search in Google Scholar

[19] Isaacs I. M., Character theory of finite groups, Corrected reprint of the 1976 original [Academic Press, New York], vol. 129. Dover Publications, Inc., New York, 1994.Search in Google Scholar

[20] Shi W. J., Bi J. X., A new characterization of the alternating groups. Southeast Asian Bull. Math., 1992, 16(1), 81–90.Search in Google Scholar

[21] Zavarnitsin A. V., Recognition of alternating groups of degrees r + 1 and r + 2 for prime r and of a group of degree 16 by the set of their element orders. Algebra Log., 2000, 39(6), 648–661, 754.Search in Google Scholar

[22] Rosser J. B., Schoenfeld L., Approximate formulas for some functions of prime numbers. Illinois J. Math., 1962, 6, 64–94.10.1215/ijm/1255631807Search in Google Scholar

[23] Pierce R., Prime factorization tool. http://www.mathsisfun.com/numbers/prime-factorization-tool.html. Accessed 17 Sep 2016.Search in Google Scholar

[24] Hall Jr., M., The theory of groups. The Macmillan Co., New York, N.Y., 1959.Search in Google Scholar

[25] Shi W. J., A new characterization of some simple groups of Lie type. In Classical groups and related topics (Beijing, 1987), vol. 82 of Contemp. Math. Amer. Math. Soc., Providence, RI, 1989, pp. 171–180.10.1090/conm/082/982286Search in Google Scholar

[26] Vasilev A. V., Grechkoseeva M. A., Mazurov V. D., Recognizability of finite simple groups by the spectrum and order. Dokl. Math., 2010, 82(2), 216–218.10.1134/S1064562410020146Search in Google Scholar
Received: 2016-11-2
Accepted: 2017-6-20
Published Online: 2017-8-19

© 2017 Yang et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

  1. Regular Articles
  2. Integrals of Frullani type and the method of brackets
  3. Regular Articles
  4. Edge of chaos in reaction diffusion CNN model
  5. Regular Articles
  6. Calculus using proximities: a mathematical approach in which students can actually prove theorems
  7. Regular Articles
  8. An investigation on hyper S-posets over ordered semihypergroups
  9. Regular Articles
  10. The Leibniz algebras whose subalgebras are ideals
  11. Regular Articles
  12. Fixed point and multidimensional fixed point theorems with applications to nonlinear matrix equations in terms of weak altering distance functions
  13. Regular Articles
  14. Matrix rank and inertia formulas in the analysis of general linear models
  15. Regular Articles
  16. The hybrid power mean of quartic Gauss sums and Kloosterman sums
  17. Regular Articles
  18. Tauberian theorems for statistically (C,1,1) summable double sequences of fuzzy numbers
  19. Regular Articles
  20. Some properties of graded comultiplication modules
  21. Regular Articles
  22. The characterizations of upper approximation operators based on special coverings
  23. Regular Articles
  24. Bi-integrable and tri-integrable couplings of a soliton hierarchy associated with SO(4)
  25. Regular Articles
  26. Dynamics for a discrete competition and cooperation model of two enterprises with multiple delays and feedback controls
  27. Regular Articles
  28. A new view of relationship between atomic posets and complete (algebraic) lattices
  29. Regular Articles
  30. A class of extensions of Restricted (s, t)-Wythoff’s game
  31. Regular Articles
  32. New bounds for the minimum eigenvalue of 𝓜-tensors
  33. Regular Articles
  34. Shintani and Shimura lifts of cusp forms on certain arithmetic groups and their applications
  35. Regular Articles
  36. Empirical likelihood for quantile regression models with response data missing at random
  37. Regular Articles
  38. Convex combination of analytic functions
  39. Regular Articles
  40. On the Yang-Baxter-like matrix equation for rank-two matrices
  41. Regular Articles
  42. Uniform topology on EQ-algebras
  43. Regular Articles
  44. Integrations on rings
  45. Regular Articles
  46. The quasilinear parabolic kirchhoff equation
  47. Regular Articles
  48. Avoiding rainbow 2-connected subgraphs
  49. Regular Articles
  50. On non-Hopfian groups of fractions
  51. Regular Articles
  52. Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions
  53. Regular Articles
  54. Rings in which elements are the sum of a nilpotent and a root of a fixed polynomial that commute
  55. Regular Articles
  56. Superstability of functional equations related to spherical functions
  57. Regular Articles
  58. Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52
  59. Regular Articles
  60. Weighted minimal translation surfaces in the Galilean space with density
  61. Regular Articles
  62. Complete convergence for weighted sums of pairwise independent random variables
  63. Regular Articles
  64. Binomials transformation formulae for scaled Fibonacci numbers
  65. Regular Articles
  66. Growth functions for some uniformly amenable groups
  67. Regular Articles
  68. Hopf bifurcations in a three-species food chain system with multiple delays
  69. Regular Articles
  70. Oscillation and nonoscillation of half-linear Euler type differential equations with different periodic coefficients
  71. Regular Articles
  72. Osculating curves in 4-dimensional semi-Euclidean space with index 2
  73. Regular Articles
  74. Some new facts about group 𝒢 generated by the family of convergent permutations
  75. Regular Articles
  76. lnfinitely many solutions for fractional Schrödinger equations with perturbation via variational methods
  77. Regular Articles
  78. Supersolvable orders and inductively free arrangements
  79. Regular Articles
  80. Asymptotically almost automorphic solutions of differential equations with piecewise constant argument
  81. Regular Articles
  82. Finite groups whose all second maximal subgroups are cyclic
  83. Regular Articles
  84. Semilinear systems with a multi-valued nonlinear term
  85. Regular Articles
  86. Positive solutions for Hadamard differential systems with fractional integral conditions on an unbounded domain
  87. Regular Articles
  88. Calibration and simulation of Heston model
  89. Regular Articles
  90. One kind sixth power mean of the three-term exponential sums
  91. Regular Articles
  92. Cyclic pairs and common best proximity points in uniformly convex Banach spaces
  93. Regular Articles
  94. The uniqueness of meromorphic functions in k-punctured complex plane
  95. Regular Articles
  96. Normalizers of intermediate congruence subgroups of the Hecke subgroups
  97. Regular Articles
  98. The hyperbolicity constant of infinite circulant graphs
  99. Regular Articles
  100. Scott convergence and fuzzy Scott topology on L-posets
  101. Regular Articles
  102. One sided strong laws for random variables with infinite mean
  103. Regular Articles
  104. The join of split graphs whose completely regular endomorphisms form a monoid
  105. Regular Articles
  106. A new branch and bound algorithm for minimax ratios problems
  107. Regular Articles
  108. Upper bound estimate of incomplete Cochrane sum
  109. Regular Articles
  110. Value distributions of solutions to complex linear differential equations in angular domains
  111. Regular Articles
  112. The nonlinear diffusion equation of the ideal barotropic gas through a porous medium
  113. Regular Articles
  114. The Sheffer stroke operation reducts of basic algebras
  115. Regular Articles
  116. Extensions and improvements of Sherman’s and related inequalities for n-convex functions
  117. Regular Articles
  118. Classification lattices are geometric for complete atomistic lattices
  119. Regular Articles
  120. Possible numbers of x’s in an {x, y}-matrix with a given rank
  121. Regular Articles
  122. New error bounds for linear complementarity problems of weakly chained diagonally dominant B-matrices
  123. Regular Articles
  124. Boundedness of vector-valued B-singular integral operators in Lebesgue spaces
  125. Regular Articles
  126. On the Golomb’s conjecture and Lehmer’s numbers
  127. Regular Articles
  128. Some applications of the Archimedean copulas in the proof of the almost sure central limit theorem for ordinary maxima
  129. Regular Articles
  130. Dual-stage adaptive finite-time modified function projective multi-lag combined synchronization for multiple uncertain chaotic systems
  131. Regular Articles
  132. Corrigendum to: Dual-stage adaptive finite-time modified function projective multi-lag combined synchronization for multiple uncertain chaotic systems
  133. Regular Articles
  134. Convergence and stability of generalized φ-weak contraction mapping in CAT(0) spaces
  135. Regular Articles
  136. Triple solutions for a Dirichlet boundary value problem involving a perturbed discrete p(k)-Laplacian operator
  137. Regular Articles
  138. OD-characterization of alternating groups Ap+d
  139. Regular Articles
  140. On Jordan mappings of inverse semirings
  141. Regular Articles
  142. On generalized Ehresmann semigroups
  143. Regular Articles
  144. On topological properties of spaces obtained by the double band matrix
  145. Regular Articles
  146. Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions
  147. Regular Articles
  148. Chain conditions on composite Hurwitz series rings
  149. Regular Articles
  150. Coloring subgraphs with restricted amounts of hues
  151. Regular Articles
  152. An extension of the method of brackets. Part 1
  153. Regular Articles
  154. Branch-delete-bound algorithm for globally solving quadratically constrained quadratic programs
  155. Regular Articles
  156. Strong edge geodetic problem in networks
  157. Regular Articles
  158. Ricci solitons on almost Kenmotsu 3-manifolds
  159. Regular Articles
  160. Uniqueness of meromorphic functions sharing two finite sets
  161. Regular Articles
  162. On the fourth-order linear recurrence formula related to classical Gauss sums
  163. Regular Articles
  164. Dynamical behavior for a stochastic two-species competitive model
  165. Regular Articles
  166. Two new eigenvalue localization sets for tensors and theirs applications
  167. Regular Articles
  168. κ-strong sequences and the existence of generalized independent families
  169. Regular Articles
  170. Commutators of Littlewood-Paley gκ∗ -functions on non-homogeneous metric measure spaces
  171. Regular Articles
  172. On decompositions of estimators under a general linear model with partial parameter restrictions
  173. Regular Articles
  174. Groups and monoids of Pythagorean triples connected to conics
  175. Regular Articles
  176. Hom-Lie superalgebra structures on exceptional simple Lie superalgebras of vector fields
  177. Regular Articles
  178. Numerical methods for the multiplicative partial differential equations
  179. Regular Articles
  180. Solvable Leibniz algebras with NFn⊕ Fm1 nilradical
  181. Regular Articles
  182. Evaluation of the convolution sums ∑al+bm=n lσ(l) σ(m) with ab ≤ 9
  183. Regular Articles
  184. A study on soft rough semigroups and corresponding decision making applications
  185. Regular Articles
  186. Some new inequalities of Hermite-Hadamard type for s-convex functions with applications
  187. Regular Articles
  188. Deficiency of forests
  189. Regular Articles
  190. Perfect codes in power graphs of finite groups
  191. Regular Articles
  192. A new compact finite difference quasilinearization method for nonlinear evolution partial differential equations
  193. Regular Articles
  194. Does any convex quadrilateral have circumscribed ellipses?
  195. Regular Articles
  196. The dynamic of a Lie group endomorphism
  197. Regular Articles
  198. On pairs of equations in unlike powers of primes and powers of 2
  199. Regular Articles
  200. Differential subordination and convexity criteria of integral operators
  201. Regular Articles
  202. Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem
  203. Regular Articles
  204. On θ-commutators and the corresponding non-commuting graphs
  205. Regular Articles
  206. Quasi-maximum likelihood estimator of Laplace (1, 1) for GARCH models
  207. Regular Articles
  208. Multiple and sign-changing solutions for discrete Robin boundary value problem with parameter dependence
  209. Regular Articles
  210. Fundamental relation on m-idempotent hyperrings
  211. Regular Articles
  212. A novel recursive method to reconstruct multivariate functions on the unit cube
  213. Regular Articles
  214. Nabla inequalities and permanence for a logistic integrodifferential equation on time scales
  215. Regular Articles
  216. Enumeration of spanning trees in the sequence of Dürer graphs
  217. Regular Articles
  218. Quotient of information matrices in comparison of linear experiments for quadratic estimation
  219. Regular Articles
  220. Fourier series of functions involving higher-order ordered Bell polynomials
  221. Regular Articles
  222. Simple modules over Auslander regular rings
  223. Regular Articles
  224. Weighted multilinear p-adic Hardy operators and commutators
  225. Regular Articles
  226. Guaranteed cost finite-time control of positive switched nonlinear systems with D-perturbation
  227. Regular Articles
  228. A modified quasi-boundary value method for an abstract ill-posed biparabolic problem
  229. Regular Articles
  230. Extended Riemann-Liouville type fractional derivative operator with applications
  231. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  232. The algebraic size of the family of injective operators
  233. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  234. The history of a general criterium on spaceability
  235. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  236. On sequences not enjoying Schur’s property
  237. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  238. A hierarchy in the family of real surjective functions
  239. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  240. Dynamics of multivalued linear operators
  241. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  242. Linear dynamics of semigroups generated by differential operators
  243. Special Issue on Recent Developments in Differential Equations
  244. Isomorphism theorems for some parabolic initial-boundary value problems in Hörmander spaces
  245. Special Issue on Recent Developments in Differential Equations
  246. Determination of a diffusion coefficient in a quasilinear parabolic equation
  247. Special Issue on Recent Developments in Differential Equations
  248. Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables
  249. Special Issue on Recent Developments in Differential Equations
  250. A nonlinear plate control without linearization
  251. Special Issue on Recent Developments in Differential Equations
  252. Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations
  253. Special Issue on Recent Developments in Differential Equations
  254. Inverse problem for a physiologically structured population model with variable-effort harvesting
  255. Special Issue on Recent Developments in Differential Equations
  256. Existence of solutions for delay evolution equations with nonlocal conditions
  257. Special Issue on Recent Developments in Differential Equations
  258. Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities
  259. Special Issue on Recent Developments in Differential Equations
  260. Coupled fixed point theorems in complete metric spaces endowed with a directed graph and application
  261. Special Issue on Recent Developments in Differential Equations
  262. Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations
  263. Special Issue on Recent Developments in Differential Equations
  264. Integro-differential systems with variable exponents of nonlinearity
  265. Special Issue on Recent Developments in Differential Equations
  266. Elliptic operators on refined Sobolev scales on vector bundles
  267. Special Issue on Recent Developments in Differential Equations
  268. Multiplicity solutions of a class fractional Schrödinger equations
  269. Special Issue on Recent Developments in Differential Equations
  270. Determining of right-hand side of higher order ultraparabolic equation
  271. Special Issue on Recent Developments in Differential Equations
  272. Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain
  273. Topical Issue on Metaheuristics - Methods and Applications
  274. Learnheuristics: hybridizing metaheuristics with machine learning for optimization with dynamic inputs
  275. Topical Issue on Metaheuristics - Methods and Applications
  276. Nature–inspired metaheuristic algorithms to find near–OGR sequences for WDM channel allocation and their performance comparison
  277. Topical Issue on Cyber-security Mathematics
  278. Monomial codes seen as invariant subspaces
  279. Topical Issue on Cyber-security Mathematics
  280. Expert knowledge and data analysis for detecting advanced persistent threats
  281. Topical Issue on Cyber-security Mathematics
  282. Feedback equivalence of convolutional codes over finite rings
https://doi.org/10.1515/math-2017-0092
Keywords for this article
Order component; Element order; Alternating group; Degree pattern; Prime graph; Simple group
Creative Commons
BY-NC-ND 4.0

Articles in the same Issue

  1. Regular Articles
  2. Integrals of Frullani type and the method of brackets
  3. Regular Articles
  4. Edge of chaos in reaction diffusion CNN model
  5. Regular Articles
  6. Calculus using proximities: a mathematical approach in which students can actually prove theorems
  7. Regular Articles
  8. An investigation on hyper S-posets over ordered semihypergroups
  9. Regular Articles
  10. The Leibniz algebras whose subalgebras are ideals
  11. Regular Articles
  12. Fixed point and multidimensional fixed point theorems with applications to nonlinear matrix equations in terms of weak altering distance functions
  13. Regular Articles
  14. Matrix rank and inertia formulas in the analysis of general linear models
  15. Regular Articles
  16. The hybrid power mean of quartic Gauss sums and Kloosterman sums
  17. Regular Articles
  18. Tauberian theorems for statistically (C,1,1) summable double sequences of fuzzy numbers
  19. Regular Articles
  20. Some properties of graded comultiplication modules
  21. Regular Articles
  22. The characterizations of upper approximation operators based on special coverings
  23. Regular Articles
  24. Bi-integrable and tri-integrable couplings of a soliton hierarchy associated with SO(4)
  25. Regular Articles
  26. Dynamics for a discrete competition and cooperation model of two enterprises with multiple delays and feedback controls
  27. Regular Articles
  28. A new view of relationship between atomic posets and complete (algebraic) lattices
  29. Regular Articles
  30. A class of extensions of Restricted (s, t)-Wythoff’s game
  31. Regular Articles
  32. New bounds for the minimum eigenvalue of 𝓜-tensors
  33. Regular Articles
  34. Shintani and Shimura lifts of cusp forms on certain arithmetic groups and their applications
  35. Regular Articles
  36. Empirical likelihood for quantile regression models with response data missing at random
  37. Regular Articles
  38. Convex combination of analytic functions
  39. Regular Articles
  40. On the Yang-Baxter-like matrix equation for rank-two matrices
  41. Regular Articles
  42. Uniform topology on EQ-algebras
  43. Regular Articles
  44. Integrations on rings
  45. Regular Articles
  46. The quasilinear parabolic kirchhoff equation
  47. Regular Articles
  48. Avoiding rainbow 2-connected subgraphs
  49. Regular Articles
  50. On non-Hopfian groups of fractions
  51. Regular Articles
  52. Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions
  53. Regular Articles
  54. Rings in which elements are the sum of a nilpotent and a root of a fixed polynomial that commute
  55. Regular Articles
  56. Superstability of functional equations related to spherical functions
  57. Regular Articles
  58. Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52
  59. Regular Articles
  60. Weighted minimal translation surfaces in the Galilean space with density
  61. Regular Articles
  62. Complete convergence for weighted sums of pairwise independent random variables
  63. Regular Articles
  64. Binomials transformation formulae for scaled Fibonacci numbers
  65. Regular Articles
  66. Growth functions for some uniformly amenable groups
  67. Regular Articles
  68. Hopf bifurcations in a three-species food chain system with multiple delays
  69. Regular Articles
  70. Oscillation and nonoscillation of half-linear Euler type differential equations with different periodic coefficients
  71. Regular Articles
  72. Osculating curves in 4-dimensional semi-Euclidean space with index 2
  73. Regular Articles
  74. Some new facts about group 𝒢 generated by the family of convergent permutations
  75. Regular Articles
  76. lnfinitely many solutions for fractional Schrödinger equations with perturbation via variational methods
  77. Regular Articles
  78. Supersolvable orders and inductively free arrangements
  79. Regular Articles
  80. Asymptotically almost automorphic solutions of differential equations with piecewise constant argument
  81. Regular Articles
  82. Finite groups whose all second maximal subgroups are cyclic
  83. Regular Articles
  84. Semilinear systems with a multi-valued nonlinear term
  85. Regular Articles
  86. Positive solutions for Hadamard differential systems with fractional integral conditions on an unbounded domain
  87. Regular Articles
  88. Calibration and simulation of Heston model
  89. Regular Articles
  90. One kind sixth power mean of the three-term exponential sums
  91. Regular Articles
  92. Cyclic pairs and common best proximity points in uniformly convex Banach spaces
  93. Regular Articles
  94. The uniqueness of meromorphic functions in k-punctured complex plane
  95. Regular Articles
  96. Normalizers of intermediate congruence subgroups of the Hecke subgroups
  97. Regular Articles
  98. The hyperbolicity constant of infinite circulant graphs
  99. Regular Articles
  100. Scott convergence and fuzzy Scott topology on L-posets
  101. Regular Articles
  102. One sided strong laws for random variables with infinite mean
  103. Regular Articles
  104. The join of split graphs whose completely regular endomorphisms form a monoid
  105. Regular Articles
  106. A new branch and bound algorithm for minimax ratios problems
  107. Regular Articles
  108. Upper bound estimate of incomplete Cochrane sum
  109. Regular Articles
  110. Value distributions of solutions to complex linear differential equations in angular domains
  111. Regular Articles
  112. The nonlinear diffusion equation of the ideal barotropic gas through a porous medium
  113. Regular Articles
  114. The Sheffer stroke operation reducts of basic algebras
  115. Regular Articles
  116. Extensions and improvements of Sherman’s and related inequalities for n-convex functions
  117. Regular Articles
  118. Classification lattices are geometric for complete atomistic lattices
  119. Regular Articles
  120. Possible numbers of x’s in an {x, y}-matrix with a given rank
  121. Regular Articles
  122. New error bounds for linear complementarity problems of weakly chained diagonally dominant B-matrices
  123. Regular Articles
  124. Boundedness of vector-valued B-singular integral operators in Lebesgue spaces
  125. Regular Articles
  126. On the Golomb’s conjecture and Lehmer’s numbers
  127. Regular Articles
  128. Some applications of the Archimedean copulas in the proof of the almost sure central limit theorem for ordinary maxima
  129. Regular Articles
  130. Dual-stage adaptive finite-time modified function projective multi-lag combined synchronization for multiple uncertain chaotic systems
  131. Regular Articles
  132. Corrigendum to: Dual-stage adaptive finite-time modified function projective multi-lag combined synchronization for multiple uncertain chaotic systems
  133. Regular Articles
  134. Convergence and stability of generalized φ-weak contraction mapping in CAT(0) spaces
  135. Regular Articles
  136. Triple solutions for a Dirichlet boundary value problem involving a perturbed discrete p(k)-Laplacian operator
  137. Regular Articles
  138. OD-characterization of alternating groups Ap+d
  139. Regular Articles
  140. On Jordan mappings of inverse semirings
  141. Regular Articles
  142. On generalized Ehresmann semigroups
  143. Regular Articles
  144. On topological properties of spaces obtained by the double band matrix
  145. Regular Articles
  146. Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions
  147. Regular Articles
  148. Chain conditions on composite Hurwitz series rings
  149. Regular Articles
  150. Coloring subgraphs with restricted amounts of hues
  151. Regular Articles
  152. An extension of the method of brackets. Part 1
  153. Regular Articles
  154. Branch-delete-bound algorithm for globally solving quadratically constrained quadratic programs
  155. Regular Articles
  156. Strong edge geodetic problem in networks
  157. Regular Articles
  158. Ricci solitons on almost Kenmotsu 3-manifolds
  159. Regular Articles
  160. Uniqueness of meromorphic functions sharing two finite sets
  161. Regular Articles
  162. On the fourth-order linear recurrence formula related to classical Gauss sums
  163. Regular Articles
  164. Dynamical behavior for a stochastic two-species competitive model
  165. Regular Articles
  166. Two new eigenvalue localization sets for tensors and theirs applications
  167. Regular Articles
  168. κ-strong sequences and the existence of generalized independent families
  169. Regular Articles
  170. Commutators of Littlewood-Paley gκ∗ -functions on non-homogeneous metric measure spaces
  171. Regular Articles
  172. On decompositions of estimators under a general linear model with partial parameter restrictions
  173. Regular Articles
  174. Groups and monoids of Pythagorean triples connected to conics
  175. Regular Articles
  176. Hom-Lie superalgebra structures on exceptional simple Lie superalgebras of vector fields
  177. Regular Articles
  178. Numerical methods for the multiplicative partial differential equations
  179. Regular Articles
  180. Solvable Leibniz algebras with NFn⊕ Fm1 nilradical
  181. Regular Articles
  182. Evaluation of the convolution sums ∑al+bm=n lσ(l) σ(m) with ab ≤ 9
  183. Regular Articles
  184. A study on soft rough semigroups and corresponding decision making applications
  185. Regular Articles
  186. Some new inequalities of Hermite-Hadamard type for s-convex functions with applications
  187. Regular Articles
  188. Deficiency of forests
  189. Regular Articles
  190. Perfect codes in power graphs of finite groups
  191. Regular Articles
  192. A new compact finite difference quasilinearization method for nonlinear evolution partial differential equations
  193. Regular Articles
  194. Does any convex quadrilateral have circumscribed ellipses?
  195. Regular Articles
  196. The dynamic of a Lie group endomorphism
  197. Regular Articles
  198. On pairs of equations in unlike powers of primes and powers of 2
  199. Regular Articles
  200. Differential subordination and convexity criteria of integral operators
  201. Regular Articles
  202. Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem
  203. Regular Articles
  204. On θ-commutators and the corresponding non-commuting graphs
  205. Regular Articles
  206. Quasi-maximum likelihood estimator of Laplace (1, 1) for GARCH models
  207. Regular Articles
  208. Multiple and sign-changing solutions for discrete Robin boundary value problem with parameter dependence
  209. Regular Articles
  210. Fundamental relation on m-idempotent hyperrings
  211. Regular Articles
  212. A novel recursive method to reconstruct multivariate functions on the unit cube
  213. Regular Articles
  214. Nabla inequalities and permanence for a logistic integrodifferential equation on time scales
  215. Regular Articles
  216. Enumeration of spanning trees in the sequence of Dürer graphs
  217. Regular Articles
  218. Quotient of information matrices in comparison of linear experiments for quadratic estimation
  219. Regular Articles
  220. Fourier series of functions involving higher-order ordered Bell polynomials
  221. Regular Articles
  222. Simple modules over Auslander regular rings
  223. Regular Articles
  224. Weighted multilinear p-adic Hardy operators and commutators
  225. Regular Articles
  226. Guaranteed cost finite-time control of positive switched nonlinear systems with D-perturbation
  227. Regular Articles
  228. A modified quasi-boundary value method for an abstract ill-posed biparabolic problem
  229. Regular Articles
  230. Extended Riemann-Liouville type fractional derivative operator with applications
  231. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  232. The algebraic size of the family of injective operators
  233. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  234. The history of a general criterium on spaceability
  235. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  236. On sequences not enjoying Schur’s property
  237. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  238. A hierarchy in the family of real surjective functions
  239. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  240. Dynamics of multivalued linear operators
  241. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  242. Linear dynamics of semigroups generated by differential operators
  243. Special Issue on Recent Developments in Differential Equations
  244. Isomorphism theorems for some parabolic initial-boundary value problems in Hörmander spaces
  245. Special Issue on Recent Developments in Differential Equations
  246. Determination of a diffusion coefficient in a quasilinear parabolic equation
  247. Special Issue on Recent Developments in Differential Equations
  248. Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables
  249. Special Issue on Recent Developments in Differential Equations
  250. A nonlinear plate control without linearization
  251. Special Issue on Recent Developments in Differential Equations
  252. Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations
  253. Special Issue on Recent Developments in Differential Equations
  254. Inverse problem for a physiologically structured population model with variable-effort harvesting
  255. Special Issue on Recent Developments in Differential Equations
  256. Existence of solutions for delay evolution equations with nonlocal conditions
  257. Special Issue on Recent Developments in Differential Equations
  258. Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities
  259. Special Issue on Recent Developments in Differential Equations
  260. Coupled fixed point theorems in complete metric spaces endowed with a directed graph and application
  261. Special Issue on Recent Developments in Differential Equations
  262. Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations
  263. Special Issue on Recent Developments in Differential Equations
  264. Integro-differential systems with variable exponents of nonlinearity
  265. Special Issue on Recent Developments in Differential Equations
  266. Elliptic operators on refined Sobolev scales on vector bundles
  267. Special Issue on Recent Developments in Differential Equations
  268. Multiplicity solutions of a class fractional Schrödinger equations
  269. Special Issue on Recent Developments in Differential Equations
  270. Determining of right-hand side of higher order ultraparabolic equation
  271. Special Issue on Recent Developments in Differential Equations
  272. Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain
  273. Topical Issue on Metaheuristics - Methods and Applications
  274. Learnheuristics: hybridizing metaheuristics with machine learning for optimization with dynamic inputs
  275. Topical Issue on Metaheuristics - Methods and Applications
  276. Nature–inspired metaheuristic algorithms to find near–OGR sequences for WDM channel allocation and their performance comparison
  277. Topical Issue on Cyber-security Mathematics
  278. Monomial codes seen as invariant subspaces
  279. Topical Issue on Cyber-security Mathematics
  280. Expert knowledge and data analysis for detecting advanced persistent threats
  281. Topical Issue on Cyber-security Mathematics
  282. Feedback equivalence of convolutional codes over finite rings
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 13.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2017-0092/html
Scroll to top button