Blocks of small defect in alternating groups and squares of Brauer character degrees

Xiaoyou Chen; James P. Cossey; Mark L. Lewis; Hung P. Tong-Viet

doi:10.1515/jgth-2017-0025

Enjoy 40% off

academic books on De Gruyter Brill *

Article Publicly Available

Blocks of small defect in alternating groups and squares of Brauer character degrees

Xiaoyou Chen , James P. Cossey , Mark L. Lewis and Hung P. Tong-Viet

Published/Copyright: July 18, 2017

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Journal of Group Theory Volume 20 Issue 6

Abstract

Let p be a prime. We show that other than a few exceptions, alternating groups will have p-blocks with small defect for p equal to 2 or 3. Using this result, we prove that a finite group G has a normal Sylow p-subgroup P and G/P is nilpotent if and only if φ⁢(1)2 divides |G:ker(φ)| for every irreducible Brauer character φ of G.

1 Introduction

In this note, all groups are finite. We will write Irr⁢(G) for the set of ordinary irreducible characters of G and IBr⁡(G) for the set of irreducible Brauer characters of G for a given prime p. In [8], the third author along with Stephen Gagola, Jr. proved that a group G is nilpotent if and only if χ⁢(1)2 divides |G:ker(χ)| for all χ∈Irr⁢(G). We now ask whether a similar result holds for Brauer characters.

First, we must recall the definition of ker⁢(φ) from [18] for a Brauer character φ. Let p be a prime and let F be an algebraically closed field of characteristic p. If φ∈IBr⁢(G), then ker⁢(φ)=ker⁢(𝒳) where 𝒳 is an F-representation affording φ. (It is not difficult to see that ker⁢(φ) is well-defined, see [18, p. 39].) We find a necessary and sufficient condition for a group G to have φ⁢(1)2 divides |G:ker(φ)| for all φ∈IBr⁢(G).

Theorem 1.

Let p be a prime and let G be a group. Then φ⁢(1)2 divides |G:ker(φ)| for all φ∈IBr⁢(G) if and only if G has a normal Sylow p-subgroup P and G/P is nilpotent.

We first prove the ‘if’ part of Theorem 1 in Lemma 3.4. The proof is very similar to the proof for ordinary characters and takes relatively little additional work. So, we will focus on the ‘only if’ part. Now suppose that φ⁢(1)2 divides |G:ker(φ)| for all φ∈IBr⁢(G). If G has a normal Sylow p-subgroup P, then it is easy to deduce from the main result in [8] that G/P is nilpotent. Therefore, we will need to show that G has a normal Sylow p-subgroup.

To prove this result, assume that the Sylow p-subgroup P of G is not normal in G. We will consider the p-solvable and non-p-solvable cases separately. For the p-solvable case, we will make use of the Bπ-characters defined in [11] and the large orbit result in [14] along with recycling some of the ideas from [8]. For the non-p-solvable case, we will compare the r-parts, for various primes r, of the degrees of the Brauer characters φ and that of |G:ker(φ)| for φ∈IBr⁡(G). We will separate the proof into two cases depending on whether a certain nonabelian composition factor S of the group possesses an ordinary irreducible character of p-defect zero or not. If the former case occurs, we can construct a Brauer character of G whose degree has a large p-part, which leads quickly to a contradiction.

When S does not have an ordinary irreducible character of p-defect zero, we know from [9, Corollary 1] that p=2 or 3 and S is isomorphic to a handful of sporadic simple groups or an alternating group Alt⁢(n) for some integer n≥7. The sporadic simple groups cases can be handled using GAP [24].

For the alternating groups, when p=2 or 3, we know that ‘most’ of the groups Alt⁢(n) have no p-blocks of defect zero. However, we will show that Alt⁢(n) possesses a p-block B whose defect is not too large in comparison with the exponent of the Sylow p-subgroup of Alt⁢(n). We then use the p-parts of the degrees of the Brauer characters in B to obtain a contradiction. Thus the following is key in our proof of Theorem 1 in the non-p-solvable case. This result will certainly be very useful in the study of the p-parts of ordinary and Brauer character degrees in finite groups.

Theorem 2.

Let n≥5 be an integer, let p=2 or 3 and let pa=|Alt⁢(n)|p. Then Alt⁢(n) has a p-block B of defect d=d⁢(B) such that d≤(a-1)/2 for p=3, and d≤(a-2)/2 for p=2 with the following exceptions:

n=7 if p=3.
n=7,9,11,13,22,24 and 26 if p=2.

This is related to a question posed in [6] by Espuelas and Navarro, where they asked whether a finite group G with 𝐎p⁢(G)=1 and |G|p=pa, where p≥5 is a prime, contains a block of defect less than [a/2].

Our notation for Brauer characters is standard and follows [18]. For the organization of this paper, we will prove Theorem 2 in Section 2 and Theorem 1 will be proved in Section 3.

2 Alternating groups

We begin with some background material. (See [15] for more details.) Recall that the ordinary irreducible characters of Sym⁢(n) are indexed by the partitions of n. Each partition λ has an associated Young diagram, and each box in the Young diagram has an associated hook length: for the box in row i, column j, the hook length is the number of boxes in the Young diagram directly to the right of (i,j), plus the number of boxes in the Young diagram directly below (i,j), plus one (for the node (i,j)). The rim hook corresponding to a box is the projection of the boxes of the hook to the outer edge of the Young diagram. In other words, the rim hook corresponding to the box (i,j) is the set of boxes starting at the end of column j, and going to the end of row i, along the edge of the Young diagram of λ. The number of boxes in the rim hook corresponding to (i,j) is necessarily equal to the hook length of that box.

A p-core is a partition that has no rim hooks of length divisible by p. For each partition λ of n, we can compute the associated p-core λ^ by successively removing all the rim p-hooks (and recomputing the hook lengths of the remaining boxes each time we remove a rim p-hook) from the Young diagram of λ. It can be shown that the end result is independent of the order the hooks are removed. Brauer [3] and Robinson [21] proved that two characters χλ and χμ are in the same p-block of Sym⁢(n) if and only if λ and μ have the same p-core.

Note that in removing rim p-hooks to get the core of λ, the number r of removed boxes is necessarily a multiple of p, and we let w=r/p be the weight of λ. Characters of Sym⁢(n) in the same block necessarily have the same weight. It is known (see [19]) that the defect group of a block B of weight w of Sym⁢(n) is a Sylow p-subgroup of Sym⁢(p⁢w).

We set some notation. For a prime p and a positive integer m, we let νp⁢(m)=a, where m=pa⁢b and b is a positive integer not divisible by p. We write ap,n=νp⁢(|Alt⁢(n)|) and sp,n=νp⁢(|Sym⁢(n)|). Of course, for odd p we have ap,n=sp,n and a2,n=s2,n-1. Recall (see [20]) that the p-blocks of Sym⁢(n) correspond to the p-blocks of Alt⁢(n). For a p-block B, we write dB,n to be the defect of the block of Alt⁢(n), and eB,n to be the defect of the block of Sym⁢(n). For odd p, we know dB,n=eB,n. However, for p=2 we know that dB,n=eB,n-1 if eB,n≥1, and if eB,n=0, then there are two blocks of Alt⁢(n) associated to B, each of which having defect zero. Thus for p=3 our desired inequality is equivalent to 2⁢eB,n+1≤sp,n. For p=2, our desired inequality is dB,n≤(a2,n-2)/2, which is equivalent (for blocks of positive defect) to eB,n-1≤(s2,n-1-2)/2, which also reduces to 2⁢eB,n+1≤s2,n.

Consequently, Theorem 2 follows from the following results on the symmetric groups.

Theorem 2.1.

For all values of n≥5 except n=7,9,11,13,22,24,26, the symmetric group Sym⁢(n) has a 2-block B of defect d such that 2⁢d+1≤s2,n. Moreover, for all values of n≥5 except n=7, Sym⁢(n) has a 3-block B of defect d such that 2⁢d+1≤s3,n.

We will need the following easy lemma.

Lemma 2.2.

Fix a prime p. For a symmetric group Sym⁢(m), let

sp,m=νp⁢(|Sym⁢(m)|).

Then there is a function fp:N→N such that if n≥fp⁢(w), then

2⁢sp,p⁢w+1≤sp,n.

In particular, if B is a block of Sym⁢(n) of weight w and defect d, then

2⁢d+1≤sp,n

if n≥fp⁢(w).

Proof.

The first statement is obvious by comparing p-parts of the orders of the symmetric groups, and the second statement follows from the first by recalling that a block of Sym⁢(n) of weight w has defect group isomorphic to a Sylow p-subgroup of Sym⁢(p⁢w) (see [19]). ∎

Of course, we have not uniquely defined fp above. But by computing sp,p⁢w and sp,n for small values of w (for p=2,3) it can be easily seen that we may take

f3⁢(1)=9,f3⁢(2)=12,f3(3) =21,f3⁢(4)=27,

f3⁢(5)=27,f3⁢(6)=36,f3(7) =42,f3⁢(8)=45.

Moreover, it is easily seen that

f2⁢(1)=4,f2⁢(2)=8,f2(3) =12,f2⁢(4)=16,f2⁢(5)=20,

f2⁢(6)=24,f2⁢(7)=26,f2(8) =32,f2⁢(9)=36,f2⁢(10)=40,

f2⁢(11)=42,f2⁢(12)=48,f2(13) =50,f2⁢(14)=60,f2⁢(15)=62,

f2⁢(16)=68,f2⁢(17)=70.

2.1 The case for p=3

We first generate some (in fact, all, though we will not need this) 3-cores. We note that the 3-cores and their sizes are well known. Let j and k be positive integers. Then we set αj,k to be the partition

(k+2⁢j,k+2⁢j-2,k+2⁢j-4,…,k+4,k+2,k,k,k-1,k-1,…,2,2,1,1).

In addition, we let βj,k be the partition

(k+2⁢j+1,k+2⁢j-1,k+2⁢j-3,…,k+3,k+1,k,k,k-1,k-1,…,2,2,1,1).

It is easily seen that both αj,k and βj,k are 3-cores. Moreover, αj,k is a partition of j⁢k+j⁢(j+1)+k⁢(k+1), and thus α0,k is a partition of k2+k. We also see that α1,k is a partition of k2+2⁢k+2, and β0,k is a partition of k2+2⁢k+1.

Notice that |α0,k|≡0mod3 if and only if k is congruent to 0 or 2mod3. Similarly, |α1,k|≡2mod3 if and only if k is congruent to 0 or 1mod3. And finally, |β0,k|≡1mod3 if and only if k is congruent to 0 or 1mod3.

Lemma 2.3.

Let n≥90. Then there is a 3-block B of Sym⁢(n) of defect d such that 2⁢d+1≤s3,n.

Proof.

We consider three cases.

Case 1. Assume n≡0mod3. Let m be the largest integer such that m2+m≤n. If m2+m is divisible by 3, set k=m. If m2+m is not divisible by 3, then (m-1)2+(m-1) is necessarily divisible by 3, and in this case we take k=m-1. By assumption, then, we have that n-(k2+k) is divisible by 3, and thus Sym⁢(n) has a 3-block B with core partition α0,k. Note we certainly have

k2+k≤n<(k+2)2+(k+2),

and thus

n-(k2+k)<(k+2)2+(k+2)-(k2+k)=4⁢k+6.

Thus any partition of n with core α0,k has 3-weight w≤(4⁢k+5)/3. Therefore the defect group P of such a block is contained in a Sylow 3-subgroup of Sym⁢(4⁢k+5). Thus

Sym⁢(4⁢k+5)×Sym⁢(4⁢k+5)×Sym⁢(3)

contains a copy of

P×P×ℤ3.

If k≥9, then k≥(7+101)/2, thus k2-7⁢k-13≥0, and so

2⁢(4⁢k+5)+3=8⁢k+13≤k2+k≤n.

Therefore

Sym⁢(4⁢k+5)×Sym⁢(4⁢k+5)×Sym⁢(3)

is contained in Sym⁢(n) if k≥9, i.e. n≥(92+9)=90, and hence 3⁢|P|2≤3s3,n. Thus 2⁢d+1≤s3,n, where d is the defect of B, and hence d≤(s3,n-1)/2, as desired, for n≥90 in this case.

The other two cases follow similarly, though we will include the details here.

Case 2. Assume n≡1mod3. We will use β0,k here, thus let m be the largest integer such that m2+2⁢m+1≤n. If m2+2⁢m+1≡1mod3, set k=m. If m2+2⁢m+1 is not equivalent to 1mod3, then (m-1)2+2⁢(m-1)+1≡1mod3, and we take k=m-1 in this case. By assumption, n-(k2+2⁢k+1) is divisible by 3, and thus Sym⁢(n) has a 3-block with core partition β0,k. Note that we certainly have k2+2⁢k+1≤n<(k+2)2+2⁢(k+2)+1, and thus

n-(k2+2⁢k+1)<(k+2)2+2⁢(k+2)+1-(k2+2⁢k+1)=4⁢k+8.

We see that any partition of n with core β0,k has 3-weight w≤(4⁢k+7)/3. Therefore the defect group P of such a block is contained in a Sylow 3-subgroup of Sym⁢(4⁢k+7). We now have that

Sym⁢(4⁢k+7)×Sym⁢(4⁢k+7)×Sym⁢(3)

contains a copy of

P×P×ℤ3.

If k≥8, then k2-6⁢k-16≥0, and hence

2⁢(4⁢k+7)+3=8⁢k+17≤k2+2⁢k+1≤n.

We have shown that

Sym⁢(4⁢k+7)×Sym⁢(4⁢k+7)×Sym⁢(3)

is contained in Sym⁢(n) if k≥8, i.e. n≥82, and therefore 3⁢|P|2≤3sn,3. Thus 2⁢d+1≤sn,3, where d is the defect of B, and thus d≤(sn,3-1)/2, as desired, for n≥82 in this case.

Case 3. Assume n≡2mod3. We will use α1,k here, thus let m be the largest integer such that m2+2⁢m+2≤n. If m2+2⁢m+2≡2mod3, set k=m. If m2+2⁢m+2 is not congruent to 2mod3, then (m-1)2+2⁢(m-1)+2≡2mod3, and in this case we set k=m-1. By assumption, n-(k2+2⁢k+2) is divisible by 3, and thus Sym⁢(n) has a 3-block with core partition α1,k. Note that we certainly have

k2+2⁢k+2≤n<(k+2)2+2⁢(k+2)+2

and so

n-(k2+2⁢k+2)<(k+2)2+2⁢(k+2)+2-(k2+2⁢k+2)=4⁢k+8.

Hence, any partition of n with core α1,k has 3-weight w≤(4⁢k+7)/3. Therefore the defect group P of such a block is contained in a Sylow 3-subgroup of Sym⁢(4⁢k+7). We deduce that

Sym⁢(4⁢k+7)×Sym⁢(4⁢k+7)×Sym⁢(3)

contains a copy of

P×P×ℤ3.

If k≥8, then k2-6⁢k-15≥0, and so

2⁢(4⁢k+7)+3=8⁢k+17≤k2+2⁢k+2≤n.

We have shown that

Sym⁢(4⁢k+7)×Sym⁢(4⁢k+7)×Sym⁢(3)

is contained in Sym⁢(n) if n≥8, and therefore 3⁢|P|2≤3s3,n. We conclude that 2⁢d+1≤sn,3, where d is the defect of B, and thus d≤(sn,3-1)/2, as desired, for n≥83 in this case. ∎

Proposition 2.4.

Theorem 2.1 holds for p=3.

Proof.

Assume that p=3. By Lemma 2.3, it remains to check that the result holds for 5≤n≤90. Of course, the result is trivially true for all values of n≥5 for which Sym⁢(n) has a block of defect zero. Similarly, if Sym⁢(n) has a block of defect 1, then the result holds for all values of n≥f3⁢(1)=9. Continuing, we see the result holds for all values of n≥f3⁢(2)=12 for which Sym⁢(n) has a block of defect two. By using the characterization of 3-cores, we see that continuing in this manner covers all values of n with 5≤n≤90 except n=7. Thus we have proven our result for p=3. ∎

2.2 The case where p=2

The case p=2 follows similarly as the p=3 case. Only here, there is only one ‘type’ of 2-core, the partition

δk=(k,k-1,k-2,…,2,1),

a partition of k⁢(k+1)/2.

Notice that k⁢(k+1)/2 is even if and only if k≡0 or 3mod4.

Proposition 2.5.

Theorem 2.1 holds for p=2.

Proof.

We let k be the largest integer such that k⁢(k+1)/2≤n and n-k⁢(k+1)/2 is even, and we note that

n-k⁢(k+1)2<(k+3)⁢(k+4)2-k⁢(k+1)2=3⁢k+6.

Thus we have n-k⁢(k+1)/2≤3⁢k+5, and therefore Sym⁢(n) has a 2-block of weight at most (3⁢k+5)/2, with defect group isomorphic to a Sylow 2-subgroup of Sym⁢(3⁢k+5). Note that if k≥13, then k2-11⁢k-26≥0, and thus

6⁢k+13≤k⁢(k+1)2≤n.

Therefore there is a copy of

Sym⁢(3⁢k+5)×Sym⁢(3⁢k+5)×Sym⁢(3)

inside of Sym⁢(n), and thus we are done for k≥13, i.e. n≥91.

As before, we handle the smaller values of n by finding the blocks of Sym⁢(n) with the smallest weight w and comparing the weight w to the values of f2⁢(w) computed above. Notice that the smallest weights for n=7,9,11,13,22,24, and 26, respectively, are w=2,3,4,5,6,7, and 8. However, these values of n are not large enough to use the function f2 defined above to cover these cases. Thus, for these values of n we do not have 2-defect groups small enough. In all other cases we are done, however. ∎

Finally, the proof of Theorem 2.1 follows from Propositions 2.4 and 2.5.

3 Squares of Brauer character degrees

We now work to prove Theorem 1. We first prove that if the squares of the degrees of the Brauer characters all divide the index of their kernels, then the group must be p-solvable. We then consider the p-solvable case where we will use the Bπ-characters due to I. M. Isaacs.

3.1 Non-p-solvable groups

We will need the following result.

Lemma 3.1.

Let S be any nonabelian simple group and let p be a prime with p dividing |S|. Then |Out⁢(S)|p<|S|p.

Proof.

If the pair (S,p) lies in the list

ℒ={(Alt⁢(7),3),(Alt⁢(7),2),(Alt⁢(11),2),(Alt⁢(13),2),(M22,2)},

then the lemma follows easily by checking [4].

Now suppose that (S,p)∉ℒ. By [7, Lemma 1.2], there exists ψ∈Irr⁢(S) such that |Aut⁡(S)|p<ψ⁢(1)p2. Since ψ⁢(1) divides |S|, we deduce that ψ⁢(1)p≤|S|p so |Aut⁡(S)|p<|S|p2 which implies that |Out⁢(S)|p<|S|p as wanted. ∎

Theorem 3.2.

Let p be a prime and let G be a group. If φ⁢(1)2 divides |G:ker(φ)| for all φ∈IBr⁡(G), then G is p-solvable.

Proof.

We may assume that G is not p-solvable. We first observe that the hypothesis of the theorem inherits to quotient groups. Suppose that G has two distinct minimal normal subgroups, say N1 and N2. Then G/Ni, i=1,2, are p-solvable by induction. Thus G is p-solvable, a contradiction. Therefore, G has a unique minimal normal subgroup N which is not abelian and p divides |N|. So we have N≅∏i=1kSi, where Si≅S and S is a nonabelian simple group with p dividing |S|. Observe that 𝐂G⁢(N)=1 so G embeds into Aut⁡(N)≅Aut⁡(S)≀Sym⁢(k). Let B=Aut(S)k∩G. Then G=B⁢H where H≅G/B is a subgroup of Sym⁢(k).

Let r be a prime not necessarily equal to p. Then we write |S|r=ra⁢(r) and |Out⁢(S)|r=rc⁢(r). Let θ∈IBr⁡(S) be non-principal with r dividing θ⁢(1) and let β=θk∈IBr⁡(N). Write θ⁢(1)r=rb⁢(r) for some integer b⁢(r)≥1 depending on r. Let φ∈IBr⁡(G) lie over β. Since G/N is p-solvable, e:=φ⁢(1)/β⁢(1) divides |G/N| by the Swan Theorem [18, Theorem 8.22]. Since N is a unique minimal normal subgroup of G and θ∈IBr⁡(S) is faithful, β=θk is faithful and so is φ. By hypothesis, we have that φ⁢(1)2⁢m=|G| for some integer m≥1. It follows that

|H|⋅|B/N|⋅|N|=m⁢φ⁢(1)2=m⁢e2⁢β⁢(1)2=m⁢e2⁢θ⁢(1)2⁢k.

Now taking the r-parts of the equation above, we obtain that

|H|r⁢|B/N|r⁢ra⁢(r)⁢k=mr⁢er2⁢r2⁢b⁢(r)⁢k.

As B/N≤Out⁢(S)k, we have

r2⁢b⁢(r)⁢k≤|Out⁢(S)|rk⋅|H|r⋅ra⁢(r)⁢k.

Furthermore, since H≤Sym⁢(k), we see that |H|r<rk/(r-1) (see [8, proof of Case 1] for instance) so

r2⁢b⁢(r)⁢k<ra⁢(r)⁢k+k/(r-1)⁢rc⁢(r)⁢k≤rk⁢(a⁢(r)+c⁢(r)+1),

which implies that

2⁢b⁢(r)<a⁢(r)+c⁢(r)+1r-1≤a⁢(r)+c⁢(r)+1

and so

($*$)2⁢b⁢(r)≤a⁢(r)+c⁢(r).

In what follows, we consider the cases when S has a p-block of defect zero or when S does not have a p-block of defect zero, separately. The first case is straight forward. For the latter, we can use GAP [24] to handle the sporadic simple groups and alternating groups of small degrees. For alternating groups of large degrees, we will need the results on block theory proved earlier.

(a) Suppose first that S has a p-block of defect zero. Then S has an irreducible ordinary character, say μ, of p-defect zero and thus θ=μ∘ is an irreducible p-Brauer character of S with θ⁢(1)p=|S|p. It follows that a⁢(p)=b⁢(p). Moreover, by Lemma 3.1, we know that c⁢(p)<a⁢(p) so c⁢(p)+1≤a⁢(p). Thus

2⁢b⁢(p)=2⁢a⁢(p)≥a⁢(p)+c⁢(p)+1

which contradicts ($*$) with r=p.

(b) Suppose that S has no characters of p-defect zero. Using the classification of p-blocks of defect zero (see, for example, [9, Corollary 2]), one of the following holds:

p=3 and S≅Suz,Co3 or Alt⁢(n) for some integer n≥7.
p=2 and S≅M12,M22,M24,J2,HS,Suz,Ru,Co1,Co3,B or Alt⁢(n) for some integer n≥7.

($**$)a≥2⁢d⁢(B)+c+1,or equivalently d⁢(B)≤12⁢(a-c-1).

(i) Using the p-parts. Assume that p=3 and S≅Suz. Then S has a 3-Brauer character θ of degree 189540=22⋅36⋅5⋅13. Also |S|3=37 and |Out⁢(S)|3=1. Thus a⁢(3)+c⁢(3)+1=7+0+1=8<2⁢b⁢(3)=12 violating ($*$).

Similarly, if p=3 and S≅Co3, then |S|3=37,|Out⁢(S)|=1 and S has an irreducible 3-Brauer character θ of degree 93312=27⋅36. Thus

a⁢(3)+c⁢(3)+1=8<2⁢b⁢(3)=12,

which is a contradiction.

Assume that (S,p)=(M12,2). Then we can choose

θ∈IBr2⁡(S)

with θ⁢(1)=24. We have that |S|2=26 and |Out⁢(S)|2=2, so

a⁢(2)+c⁢(2)+1=6+1+1=8=2⁢b⁢(2)

violating ($*$).

Assume that (S,p)=(M24,2). Then we can choose θ∈IBr2⁡(S) with

θ⁢(1)=1792=28⋅7.

We have that |S|2=210 and |Out⁢(S)|=1, so

a⁢(2)+c⁢(2)+1=10+0+1=11<16=2⁢b⁢(2)

violating ($*$).

Assume that (S,p)=(J2,2). Then we can choose θ∈IBr2⁡(S) with

θ⁢(1)=26.

We have that |S|2=27 and |Out⁢(S)|2=2, so

a⁢(2)+c⁢(2)+1=7+1+1=9<12=2⁢b⁢(2)

violating ($*$).

Assume that (S,p)=(HS,2). Then we can choose θ∈IBr2⁡(S) with

θ⁢(1)=1408=27⋅11.

We have that |S|2=29 and |Out⁢(S)|2=2, so

a⁢(2)+c⁢(2)+1=9+1+1=11<14=2⁢b⁢(2)

violating ($*$).

Assume that (S,p)=(Suz,2). Then we can choose θ∈IBr2⁡(S) with

θ⁢(1)=102400=212⋅52.

We have that |S|2=213 and |Out⁢(S)|2=2, so

a⁢(2)+c⁢(2)+1=13+1+1=15<24=2⁢b⁢(2)

violating ($*$).

Assume that (S,p)=(Ru,2). Then we can choose θ∈IBr2⁡(S) with

θ⁢(1)=8192=213.

We have that |S|2=214 and |Out⁢(S)|2=1, so

a⁢(2)+c⁢(2)+1=14+0+1=15<26=2⁢b⁢(2)

violating ($*$).

Assume that (S,p)=(Co3,2). Then we can choose θ∈IBr2⁡(S) with

θ⁢(1)=131584=29⋅257.

We have that |S|2=210 and |Out⁢(S)|2=1, so

a⁢(2)+c⁢(2)+1=10+0+1=11<18=2⁢b⁢(2)

violating ($*$).

(ii) Using the r-parts with r≠p. Suppose that (S,p)=(M22,2). Then S has an irreducible character θ∈IBr2⁡(S) with θ⁢(1)=2⋅17. We see that

|S|17=1=|Out⁢(S)|17.

So a⁢(17)+c⁢(17)+1=1<2=2⁢b⁢(17) violating ($*$).

(iii) Assume that (S,p)=(Co1,2) or (B,2). In these two cases, the 2-Brauer character tables of S are not available. Recall that a=a⁢(p), so |S|p=pa. Moreover, c=c⁢(p)=0 as |Out⁢(S)|=1.

If (S,p)=(Co1,2), then a=21 and S has a 2-block B with defect d⁢(B)=3. Similarly, if (S,p)=(B,2), then a=41 and S has a 2-block B with defect d⁢(B)=3. In both cases, we have a+c>2⁢d⁢(B)+1 and the result follows.

(iv) Assume that S=Alt⁢(7) and p=3. Then we can choose θ∈IBr3⁡(S) with θ⁢(1)=13. Then |S|13=1 and |Out⁢(S)|13=1 so

2⁢b⁢(13)=2>1=a⁢(13)+c⁢(13)+1

violating ($*$).

(v) Assume that S≅Alt⁢(n) with n>7 and p=3. In this case, c=0 since |Out⁢(S)|=2. It follows from Theorem 2 that Alt⁢(n) has a 3-block B with defect d=d⁢(B) satisfying the inequality

d≤12⁢(a-1).

Therefore, ($**$) occurs and thus this case cannot happen.

(vi) Assume that S=Alt⁢(n) with n≤13 and p=2. If S=Alt⁢(7), then we can choose θ∈IBr2⁡(S) with θ⁢(1)=2⋅7. Then |S|7=7 and |Out⁢(S)|7=1 so

2⁢b⁢(7)=2≥a⁢(7)+c⁢(7)+1

violating ($*$).

If S=Alt⁢(9), then there exists θ∈IBr2⁡(S) with θ⁢(1)=25⋅5. Moreover, we have |S|2=26 and |Out⁢(S)|2=2 so

a⁢(2)+c⁢(2)+1=8<10=2⁢b⁢(2),

a contradiction.

If S=Alt⁢(11), then there exists θ∈IBr2⁡(S) with θ⁢(1)=25⋅13. Moreover, |S|2=27 and |Out⁢(S)|2=2 so

a⁢(2)+c⁢(2)+1=9<10=2⁢b⁢(2),

a contradiction.

If S=Alt⁢(13), then there exists θ∈IBr2⁡(S) with θ⁢(1)=4224=27⋅3⋅11. Moreover, |S|2=29 and |Out⁢(S)|2=2 so

a⁢(2)+c⁢(2)+1=11<14=2⁢b⁢(2),

a contradiction.

If n is 5 or 6, then Alt⁢(n) is a simple group of Lie type and we are done by the above comments. If n∈{8,10,12}, then we are done by Theorem 2.

(vii) Assume that S≅Alt⁢(n) with n>13 and p=2. In this case, we see that c=1 and the result follows by applying Theorem 2 unless n=22,24 or 26. For these cases, we will use some 2-Brauer character degrees of Alt⁢(n) which can be found in [2, Proposition 5.2], for example.

Let λ=(n-2,2) and μ=(n-3,3). Since n≥22, we know that both λ and μ are 2-regular partitions of n. It follows from [1] that the simple modules in characteristic p, Dλ and Dμ, labelled by λ and μ respectively, remain irreducible upon restriction to Alt⁢(n) and their dimensions are given in [2, Proposition 5.2]. We have

dim⁢Dλ={12⁢(n-1)⁢(n-4)if n≡0 mod 4,12⁢n⁢(n-3)-1if n≡1 mod 4,12⁢(n-1)⁢(n-4)-1if n≡2 mod 4,12⁢n⁢(n-3)if n≡3 mod 4,

and

dim⁢Dμ={16⁢n⁢(n-2)⁢(n-7)if n≡0 mod 4,16⁢n⁢(n-1)⁢(n-5)if n≡1 mod 4,16⁢(n-1)⁢(n-2)⁢(n-6)if n≡2 mod 4,16⁢(n+1)⁢(n-1)⁢(n-6)if n≡3 mod 4.

If φ∈IBr2⁡(Alt⁢(n)) is such that φ⁢(1) is divisible by a prime r>n/2>2, then a⁢(r)=1 or 0 depending on whether r≤n or r>n and c⁢(r)=0 while b⁢(r)=1, therefore 2⁢b⁢(r)=2>a⁢(r)+c⁢(r) violating ($*$).

If n=22, then n≡2 mod 4 so

dim⁢Dλ=12⁢(22-1)⁢(22-4)-1=22⋅47.

Similarly, if n=26, then n≡2 mod 4 so

dim⁢Dλ=12⁢(26-1)⁢(26-4)-1=2⋅137.

If n=24, then n≡0 mod 4 so

dim⁢Dμ=16⁢24⁢(24-2)⁢(24-7)=23⋅11⋅17.

Thus in each case, Alt⁢(n) has a 2-Brauer character of degree divisible by a prime r>n/2 and so the result follows. The proof is now complete. ∎

3.2 p-solvable groups

In order to prove Theorem 1 under the assumption that G is p-solvable, we use the Bπ-characters of Isaacs that were initially defined in [11, Definition 5.1]. Since the definition of Bπ-characters is somewhat complicated, we do not repeat it here, but refer the interested reader to [11] or to the expository accounts in [12] and [13]. For our purposes, it is enough to know that if π is a set of primes and G is a π-separable group, then Bπ⁢(G) is a subset of Irr⁢(G). In particular, if π=p′ and G is p-solvable, then restriction to the p-regular elements of G yields a bijection between Bp′⁢(G) and IBr⁢(G). We write Go for the set of p-regular elements of G and if Θ is a character of G, then we write Θo for the restriction of Θ to Go. Hence, the map χ↦χo is the bijection from Bp′⁢(G) to IBr⁢(G). For our work, we need to establish that this map preserves kernels.

Lemma 3.3.

Let p be a prime and let G be a p-solvable group. If φ∈IBr⁢(G) and χ∈Bp′⁢(G) are such that χo=φ, then ker⁢(φ)=ker⁢(χ).

Proof.

Let N=ker⁢(φ) and M=ker⁢(χ). It follows that φ=χo is a Brauer character of G/M. Hence, φ has an F-representation 𝒳 of G/M that affords φ. Observe that N=ker⁢(𝒳)=ker⁢(φ) must contain M, so M≤N. Conversely, suppose that 𝒴 is an F-representation that affords φ. Note that N=ker⁢(𝒴). It follows that we can view φ∈IBr⁢(G/N). We know that there exists ψ∈Bp′⁢(G/N) so that ψo=φ. By [16], we showed that

Bπ⁢(G)∩Irr⁢(G/N)=Bπ⁢(G/N)

for any set of primes π. This shows that Bp′⁢(G/N)⊆Bp′⁢(G). Since there is a unique character in Bp′⁢(G) that lifts φ, we conclude that χ=ψ, and thus, N≤M. We now have M=N as desired. ∎

The next lemma will prove the ‘if’ part of Theorem 1.

Lemma 3.4.

Let p be a prime. If G has a normal Sylow p-subgroup P and G/P is nilpotent, then φ⁢(1)2 divides |G:ker(φ)| for all φ∈IBr⁢(G).

Proof.

Suppose that G has a normal Sylow p-subgroup P and G/P is nilpotent. Let φ∈IBr⁢(G). There is a character χ∈Bp′⁢(G) so that χo=φ. As P=𝐎p⁢(G), we have via [11, Corollary 5.3] that P≤ker⁢(χ). Now, G/P is a p′-group and hence χ∈Irr⁢(G/P). Applying [8, Theorem A], we have χ(1)2∣|G:ker(χ)|. Since φ⁢(1)=χ⁢(1) and ker⁢(φ)=ker⁢(χ), we obtain the desired conclusion that φ⁢(1)2 divides |G:ker(φ)|. ∎

We now prove the ‘only if’ part of Theorem 1 when G is p-solvable.

Theorem 3.5.

Let p be a prime, and let G be a p-solvable group. If φ⁢(1)2 divides |G:ker(φ)| for all φ∈IBr⁢(G), then G has a normal Sylow p-subgroup P and G/P is nilpotent.

Proof.

Suppose that φ⁢(1)2 divides |G:ker(φ)| for all φ∈IBr⁢(G). We will work by induction on |G|. Assume first that 𝐎p⁢(G)>1. It is easily seen that the hypothesis holds in G/𝐎p⁢(G), so G/𝐎p⁢(G) has a normal Sylow p-subgroup, meaning G has a normal Sylow p-subgroup. Moreover, induction (applied to G/𝐎p⁢(G)) shows that G/𝐎p⁢(G) is nilpotent.

Thus we may assume 𝐎p⁢(G)=1. If we can prove that G is a p′-group, then we will have that Irr⁢(G)=IBr⁢(G) and we will be done by [8, Theorem A]. We work to prove that G is a p′-group.

Let M be a minimal normal subgroup of G. Since 𝐎p⁢(G)=1, we know that M is a p′-subgroup of G. It is not difficult to see that G/M will satisfy the hypothesis. By induction, we see that G/M will have a normal Sylow p-subgroup N/M and G/N will be nilpotent. If K is a Sylow p-subgroup of G, then N=M⁢K. The Frattini argument implies that G=N⁢ℕG⁢(K)=M⁢K⁢ℕG⁢(K)=M⁢ℕG⁢(K).

Note that if M1 is a minimal normal subgroup of G other than M, then M1 would also be a p′-subgroup of G, so M1∩N=1, so M1 centralizes K. Applying the argument of the previous paragraph with M1 in place of M, we obtain G=M1⁢ℕG⁢(K), and since M1 centralizes K, we have G=ℕG⁢(K), and K is normal in G. Since 𝐎p⁢(G)=1, we have K=1, and G is a p′-group as desired.

Therefore, we may assume that M is the unique minimal normal subgroup of G. Assuming M is not abelian, we now mimic the argument in [8, Case 1 of the proof of Theorem A] to obtain a contradiction. We know that M is the direct product of k copies of some nonabelian simple group S. Since M is unique, we know that ℂG⁢(M)=1. This implies that G is isomorphic to a subgroup of Aut⁢(M), and Aut⁢(M) is an extension of the direct product of k copies of Aut⁢(S) by the symmetric group Sym⁢(k).

Using [8, Proposition 2.1], there is a prime q and a character σ∈Irr⁢(S) so that σ⁢(1)q=|S|q and |Out⁢(S)|q<|S|q. Since M is a p′-group and G is p-solvable, there is a character χ∈Bp′(G)∩Irr(G∣σ×⋯×σ), and observe that σ⁢(1)k divides χ⁢(1). Since σ has q-defect 0, we see that |M|q divides χ⁢(1)q. Using the same computation as in [8, Case 1 of the proof of Theorem A], we see that

|G|q<|M|q2≤χ⁢(1)q2,

and so χ⁢(1)2 does not divide |G|. If we let φ=χo∈Ip′⁢(G) (=IBr⁡(G)), then φ⁢(1)2 does not divide |G:ker(φ)|, a contradiction.

Thus, we have that M is an abelian minimal normal subgroup of G. This implies that M is a q-group for some prime q. Observe that ℕG⁢(K) will normalize ℂK⁢(M), and obviously, M normalizes ℂK⁢(M). Hence, ℂK⁢(M) is normal in G=M⁢ℕG⁢(K). Since M∩K=1 and M is the unique minimal normal subgroup of G, we conclude that ℂK⁢(M)=1. Hence, we can view Irr⁢(M) as a faithful module for K with characteristic q.

Applying Isaacs’ large orbit result [14, Theorem A], we see that there exists a character λ∈Irr⁢(M) so that |ℂK⁢(λ)|<|K|, and observe that this implies that |K:ℂK(λ)|>|K|. We see that M⁢ℂK⁢(λ) is the stabilizer of λ in N. We know that λ extends to θ∈Bp′⁢(M⁢ℂK⁢(λ)) and then γ=θN∈Bp′⁢(N). We have γ(1)=|N:MℂK(λ)|=|K:ℂK(λ)| and so, γ⁢(1)2 does not divide |K|. Let χ be an irreducible constituent of γG, and we know that χ∈Bp′⁢(G). Observe that χ⁢(1)p=γ⁢(1) and |K|=|G|p. It follows that χ⁢(1)2 does not divide |G| which is a contradiction. ∎

Now the ‘only if’ part of Theorem 1 follows from Theorems 3.2 and 3.5.

In [11], Isaacs used the Bπ-characters to extend the ideas of Brauer characters to the set of primes π in place of a single prime p for π-separable groups. Thus, we consider our question in this setting. To do this, we review some of the key points of π-theory.

We fix a set of primes π and a π-separable group G. We define the π-partial characters of G to be the restrictions of characters of G to G*, where G* denotes the set of elements of G with order only divisible by primes in π. We define Iπ⁢(G) to be those π-partial characters that cannot be written as the sum of other partial characters. In [11, Theorem A], Isaacs proves that Iπ⁢(G) forms a basis for the set of complex-valued π-class functions on G. (Note that in [11] the symbol Iπ⁢(G) is used to denote the π′-partial characters of G.) In [11, Corollary 10.2] it is shown that restriction is a bijection from Bπ⁢(G) to Iπ⁢(G). Recall that Bπ⁢(G)⊆Irr⁢(G) and note that Iπ⁢(G) plays the role for π-partial characters that IBr⁢(G) plays for Brauer characters.

Following Lemma 3.3, if φ∈Iπ⁢(G), then we define ker⁢(φ)=ker⁢(χ), where χ∈Bπ⁢(G) such that χo=φ. It now makes sense to ask whether the condition for p-solvable groups translates to a similar condition for π-separable groups. It is easy to obtain the following generalization of Lemma 3.4: Let π be a set of primes. If G has a normal Hall π-complement N and G/N is nilpotent, then φ⁢(1)2 divides |G:ker(φ)| for all φ∈Iπ⁢(G).

For the converse of this statement, we can prove the following.

Theorem 3.6.

Let π be a set of primes, and let G be a finite π-separable group. If φ⁢(1)2 divides |G:ker(φ)| for all φ∈Iπ⁢(G), then one of the following occurs:

G has a normal Hall π -complement N and G/N is nilpotent.
There is a prime p∈π and a π′-group K with a faithful module V of characteristic p so that |ℂK⁢(v)|≥|K| for all v∈V and G has a section isomorphic to VK.

In particular, the key for proving Theorem 3.5 is Isaacs’ Large Orbit Theorem stating that if K is a q-group that acts coprimely on a group V, then there exists an element v∈V so that |ℂK⁢(v)|<|K|. Recently, Halasi and Podoski have proven in [10, Corollary 1.4] that if p is a prime and K is a p′-group with a faithful module V of characteristic p, there exist v,w∈V so that ℂK⁢(v)∩ℂK⁢(w)=1. We note that this had been proved earlier for supersolvable groups by Wolf in [23], for groups of odd order by Moretó and Wolf in [17], and for solvable groups by Dolfi [5] and Vdovin in [22]. If there exists v∈V so that |ℂK⁢(v)|<|K|, then (2) does not hold. Thus, in a group as in conclusion (2), we have |ℂK⁢(v)|=|ℂK⁢(w)|=|K| and G=ℂK⁢(v)⁢ℂK⁢(w), so there exist at least two elements of v whose centralizers in K have size |K|. However, for our purposes, we need the existence of an element v∈V with |ℂK⁢(v)|<|K|. At this time, we do not see how to prove the existence of such an element, but we also do not know of any examples where this happens.

Communicated by Nigel Boston

Funding source: Henan University of Technology

Award Identifier / Grant number: 2014JCYJ14

Award Identifier / Grant number: 2016JJSB074

Award Identifier / Grant number: 26510009

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11571129

Funding statement: The first author thanks the support of China Scholarship Council, Department of Mathematical Sciences of Kent State University for its hospitality, Funds of Henan University of Technology (2014JCYJ14, 2016JJSB074, 26510009), the Project of Department of Education of Henan Province (17A110004), the Projects of Zhengzhou Municipal Bureau of Science and Technology (20150249, 20140970), and the NSFC (11571129).

Acknowledgements

The authors are grateful to the referee for the careful reading of our manuscript and for many helpful suggestions and corrections. The fourth author thanks Christine Bessenrodt for her help during the preparation of this paper.

References

[1] D. Benson, Spin modules for symmetric groups, J. Lond. Math. Soc. (2) 38 (1988), no. 2, 250–262. 10.1112/jlms/s2-38.2.250Search in Google Scholar

[2] C. Bessenrodt and H. Weber, On p-blocks of symmetric and alternating groups with all irreducible Brauer characters of prime power degree, J. Algebra 320 (2008), no. 6, 2405–2421. 10.1016/j.jalgebra.2008.04.029Search in Google Scholar

[3] R. Brauer, On a conjecture by Nakayama, Trans. Roy. Soc. Canada. Sect. III. (3) 41 (1947), 11–19. Search in Google Scholar

[4] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Oxford University Press, Oxford, 1985. Search in Google Scholar

[5] S. Dolfi, Large orbits in coprime actions of solvable groups, Trans. Amer. Math. Soc. 360 (2008), no. 1, 135–152. 10.1090/S0002-9947-07-04155-4Search in Google Scholar

[6] A. Espuelas and G. Navarro, Blocks of small defect, Proc. Amer. Math. Soc. 114 (1992), no. 4, 881–885. 10.1090/S0002-9939-1992-1070516-1Search in Google Scholar

[7] S. M. Gagola, Jr., A character theoretic condition for F⁢(G)>1, Comm. Algebra 33 (2005), no. 5, 1369–1382. 10.1081/AGB-200058414Search in Google Scholar

[8] S. M. Gagola, Jr. and M. L. Lewis, A character-theoretic condition characterizing nilpotent groups, Comm. Algebra 27 (1999), no. 3, 1053–1056. 10.1080/00927879908826480Search in Google Scholar

[9] A. Granville and K. Ono, Defect zero p-blocks for finite simple groups, Trans. Amer. Math. Soc. 348 (1996), no. 1, 331–347. 10.1090/S0002-9947-96-01481-XSearch in Google Scholar

[10] Z. Halasi and K. Podoski, Every coprime linear group admits a base of size two, Trans. Amer. Math. Soc. 368 (2016), no. 8, 5857–5887. 10.1090/tran/6544Search in Google Scholar

[11] I. M. Isaacs, Characters of π-separable groups, J. Algebra 86 (1984), no. 1, 98–128. 10.1016/0021-8693(84)90058-9Search in Google Scholar

[12] I. M. Isaacs, Characters of solvable groups, The Arcata Conference on Representations of Finite Groups (Arcata 1986), Proc. Sympos. Pure Math. 47, American Mathematical Society, Providence (1987), 103–109. 10.1090/pspum/047.1/933354Search in Google Scholar

[13] I. M. Isaacs, The π-character theory of solvable groups, J. Aust. Math. Soc. Ser. A 57 (1994), no. 1, 81–102. 10.1017/S1446788700036077Search in Google Scholar

[14] I. M. Isaacs, Large orbits in actions of nilpotent groups, Proc. Amer. Math. Soc. 127 (1999), no. 1, 45–50. 10.1090/S0002-9939-99-04584-0Search in Google Scholar

[15] G. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia Math. Appl. 16, Addison-Wesley, Reading, 1981. Search in Google Scholar

[16] M. L. Lewis, Bπ-characters and quotients, preprint (2016), https://arxiv.org/abs/1609.02029. Search in Google Scholar

[17] A. Moretó and T. R. Wolf, Orbit sizes, character degrees and Sylow subgroups, Adv. Math. 184 (2004), no. 1, 18–36. 10.1016/S0001-8708(03)00093-8Search in Google Scholar

[18] G. Navarro, Characters and Blocks of Finite Groups, London Math. Soc. Lecture Note Ser. 250, Cambridge University Press, Cambridge, 1998. 10.1017/CBO9780511526015Search in Google Scholar

[19] J. B. Olsson, Lower defect groups in symmetric groups, J. Algebra 104 (1986), no. 1, 37–56. 10.1016/0021-8693(86)90232-2Search in Google Scholar

[20] J. B. Olsson, On the p-blocks of symmetric and alternating groups and their covering groups, J. Algebra 128 (1990), no. 1, 188–213. 10.1016/0021-8693(90)90049-TSearch in Google Scholar

[21] G. D. B. Robinson, On a conjecture by Nakayama, Trans. Roy. Soc. Canada. Sect. III. (3) 41 (1947), 20–25. Search in Google Scholar

[22] E. P. Vdovin, Regular orbits of solvable linear p′-groups, Sib. Èlektron. Mat. Izv. 4 (2007), 345–360. Search in Google Scholar

[23] T. R. Wolf, Large orbits of supersolvable linear groups, J. Algebra 215 (1999), no. 1, 235–247. 10.1006/jabr.1998.7730Search in Google Scholar

[24] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.5, 2016, http://www.gap-system.org. Search in Google Scholar

Received: 2017-2-10

Revised: 2016-6-23

Published Online: 2017-7-18

Published in Print: 2017-11-1

Articles in the same Issue

https://doi.org/10.1515/jgth-2017-0025