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Generators for maximal subgroups of Conway group Co1

  • Faisal Yasin EMAIL logo , Adeel Farooq , Waqas Nazeer and Shin Min Kang EMAIL logo
Published/Copyright: April 29, 2019
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Open Mathematics
From the journal Open Mathematics Volume 17 Issue 1

Abstract

The Conway groups are the three sporadic simple groups Co1, Co2 and Co3. There are total of 22 maximal subgroups of Co1 and generators of 6 maximal subgroups are provided in web Atlas of finite simple groups. The aim of this paper is to give generators of remaining 16 maximal subgroups.

Keywords: Conway group; maximal subgroup; generators; finite group; normalizer
MSC 2010: 20D05; 20B40

1 Introduction

The Conway group Co1 is one of the 26 sporadic simple groups. The largest of the Conway groups, Co0, is the group of automorphisms of the Leech lattice Γ with respect to addition and inner product. It has order 8,315,553,613,086,720,000 [1] but it is not a simple group. The simple group Co1 of order 221.39.54.72.11.13.23 is defined as the quotient of Co0 by its center, which consists of the scalar matrices ±1 [1]. The local subgroups of Co1 are found in [2] and the maximal subgroups of Co1 in [3]. There is also a valuable discussion in [4].

The following theorem is crucial in determining the maximal subgroups of Co1:

Theorem 1.1

[3] If K is a non-Abelian characteristically simple subgroup of Co1, then NCo1(K) is contained either in a local subgroup of Co1 or in a conjugate of one of six particular groups:

  1. NA5 ≅ (A5 × J2).2,

  2. NA6 ≅ (A6 × U3(3)).2,

  3. NA7 ≅ (A7 × L2(7)).2,

  4. S(2) ≅ Co2,

  5. S(3) ≅ Co3,

  6. S(23) ≅ U6(2).S3.

Note on notation. We use A.B to denote an arbitrary extension of A by B, while A : B and A ⋅ B denote split and non-split extensions, respectively. The symbol n denotes a cyclic group of that order, while [n] denotes an arbitrary group of order n. We follow the ATLAS [5] notation for conjugacy classes. Moreover, we denote xy by y−1xy and [x, y] by x−1y−1xy.

To facilitate the computations in finite simple and almost simple groups, [6], [7] provides the representations and words for generators of most of the maximal subgroups. However, there are still some cases to deal with. A research problem “Words for maximal subgroups in sporadic groups” appears on the web page of R. A. Wilson. We pursue the work initiated by R.A. Wilson of finding words for maximal subgroups of Co1. According to [3] there are 24 conjugacy classes of maximal subgroups, but later on R.A. Wilson pointed out a few errors (in his own paper) in the list of maximal subgroups of Co1 [8] in which he mentions that the two subgroups 32.[2.36].2A4 and 32 .[23 .34].2A4 are not the maximal subgroups of Co1, so the list contains total 22 conjugacy classes of maximal subgroups. There are 22 maximal subgroups of the group Co1}. The maximal local subgroups have been determined in [3]. The Atlas of Group Representations [6] contains the words for maximal subgroups of Co1 except 16. The maximal subgroups of Co1 are given below.

  1. * (A9 × S3)

  2. * (D10 × (A5× A5).2).2

  3. * 51+2GL2(5)

  4. * 31+4 : 2.S4(3).2

  5. * 36 : 2.M12.

  6. * 32.U4(3).D8

  7. * 33+4 : 2. (S4 × S4)

  8. * 24+12.(S3 × 3S6)

  9. * 53 : (4 × A5).2

  10. * 72 : (3 × 2.S4)

  11. * 52 : 2A5

  12. * (A7 × L2(7)) : 2

  13. * (A6 × U3(3)) : 2

  14. * (A4 × G2(4)) : 2

  15. * (A5 × J2) : 2

  16. * 22+12(A8 × S3)

  17. U6(2) : S3

  18. 21+8.O8(2)

  19. Co3

  20. 211 : M24

  21. 3.Suz : 2

  22. Co2

Next we proceed to find words for 16 maximal subgroups which are marked by asterisk.

2 Methods

Most of the maximal subgroups on our list can be generated by two elements. If the group is small enough, a random search will produce the subgroup required. This method was successfully used in [9] but in Co1 the subgroups are too large to use brute force. One more focused way of generating a subgroup is by choosing a pair of conjugacy classes A and B in G such that conjugates of random elements a ∈ A, b ∈ B have a reasonable high probability of generating a conjugate of the desired subgroup.

In most of the cases we present here, even though the subgroup we wish to construct may be generated by two elements, it may be hard to tell which conjugacy classes they belong to. Even if we know a suitable pair of conjugacy classes it may be that the probability the random elements in these classes generate the desired subgroup is relatively small. In this case, we find some part of the desired subgroup, and work inside another subgroup, usually an involution centralizer, to find the rest. Once we have found a copy of the desired subgroup, we can get information regarding the generating sets.

The maximal subgroups often occur as normalizers of elementary abelian groups. So normalizers, which are crux of the matter here, were mostly computed by methods given in [10] and [11].

The generators of subgroups, wherever possible, have been obtained from [6].

The subgroups can be identified by determining order, composition series and orbit sizes in several permutation representations. Moreover, comparing our result with the list of maximal subgroups in [5] we find that there is only one possibility of the subgroup.

3 Main Results

We have extensively used the information given in [5], [3] and Atlas of finite group representations [6]. We use GAP [12] for group theoratic calculations. Throughout this paper a and b are the standard generators of Co1 in permutation representation on 98280 points available at [6].

3.1 Construction of (A9 × S3) inside Co1

The required maximal subgroup is the normalizer of an element of class 3D. Here we use power maps to find the representative of class 3D to say that a1 is the element of said class given by a1 = ((ba)2b)4 . Now the normalizer of a1 inside Co1 gives us the required maximal subgroup. The normalizer can be computed by the technique given in [10] and the programs given in [11]. Before computing the normalizer we will give some random words of Co1 which will be used later. These words are given below:

b1=(ab)2ba,b2=aba,b3=abb4=(ab)2,b5=(ab)2a,b6=(ab)2b.

Consider the group generated by a1 and b say H2 = < a1, b >, then compute the normalizer of a1 inside H2 . From here we get the partial normalizer of a1 inside Co1. Before computing the partial normalizer we will give some random elements of H2 which will facilitates our computations. These elements are given by:

c1=a1b,c2=a1ba1,c3=a1ba1bb,c4=a1ba1bba1,c5=a1ba1bba1b.

The words for partial normalizer are given below:

k1=b2b5b2b52b2b56b2b56b2b5,k2=b2b52b2b52b2b53b2b57b2b5,k3=a1c1a1c1a1c15a1c1a1c1,k4=a1c1a1c13a1c12a1c1a1c15,k5=a1c1a1c13a1c13a1c14a1c14,k6=a1c1a1c15a1c13a1c15a1c1.

Next we will find an involution inside the above partial normalizer. This involution is given by d1=k13, then find the centralizer of d1 inside Co1 by using the method given by J.Bray [13]. The generators of the centralizer of d1 Inside Co1 are given by:

d2=a[(d1,a)]2,d3=[(d1,b)]2,d4=ab[(d1,ab)]10,d5=aba[(d1,aba)]7,d6=[(d1,abab)]7,d7=[(d1,ababa)]7,d8=d2d4,d9=d2d5,d10=d2d6,d11=d2d7.

H3 = < d2, d3, d4, d5, d6, d7 >.

The words for the normalizer of a1 inside the above centralizer (H3) are given below:

k7=d2d82d2d85d2d83d2d85d2d86,k8=d3d94d3d97d3d9d3d94d3d9k9=d3d97d3d93d3d94d3d94d3d9,k10=d5d112d5d112d5d112d5d112d5d112.

By looking at the [5] we see that these words generate only partial normalizer so we repeat the above process untill we find the complete normalizer. Now we give some other elements of Co1 which will be used in further computations.

m1=(b1b2b3b42b52b63)4,m2=(b1b2b3b42b53b62)6,m3=(b1b2b3b43b5b62)3m4=(b1b2b3b43b5b62)6,m5=(b1b2b3b43b5b63)3.

Consider the group generated by a1 and m5 say H4 = < a1, m5 >, then compute the normalizer of a1 inside H4. From here we get the partial normalizer of a1 inside Co1. Before computing the partial normalizer we will give some random elements of H4. These elements are given by:

n1=a1m5,n2=a1m5m5,n3=a1m5m5a1,n4=a1m5m5a1m5,n5=a1m5m5a1m5a1,n6=a1m5m5a1m5a1m5,n7=a1m5m5a1m5a1m5m5.

The word for the normalizer of a1 inside H4 is given by:

k11=a1n74a1n74a1n74a1n74a1n74.

Now by combining the words for the normalizer of a1 inside H2, H3 and H4 we get the required complete normalizer given by k2, k7 and k11. The generators for (A9 × S3) are k2 and k7k11.

3.2 Construction of (D10 × (A5 × A5).2).2 inside Co1

The required maximal subgroup is the normalizer of an element of class 5B. Here we first find an element of class 5B and then find the normalizer of that element inside Co1, which gives us the required maximal subgroup. The element of class 5B can be calculated by using the power maps. The normalizer can be computed by the technique given in [10] and the programs given in [11] will facilitate us in computing the normalizer. Before computing the normalizer we will give some elements of Co1 which will be used later. These elements are given below:

b1=ababba,b2=aba,b3=ab,b4=abab,b5=ababa,b6=ababb,b7=ababbab,b12=babbababab.

The element of class 5C is given by c = (b1b12)2. Next we will give the strategy of finding the normalizer of c inside Co1. Consider the group generated by c and a say H2 = < c, a >, then compute the normalizer of c inside H2. From here we get the partial normalizer of c inside Co1. Before computing the partial normalizer we will give some random elements of H2 which will facilitates us in computations.

c1=ca,c2=cac,c6=caccac,c7=accaca,c8=accacaca,c9=accacacac.

The words for normalizer of c inside H2 are given by:

k1=cc1cc13cc13cc13cc12,k2=cc13cc16cc12cc16cc13,k3=cc14cc17cc16cc17cc14,k4=ac66ac63ac63ac65ac64,k5=ac9ac9ac97ac94ac96.

Here we will give some more elements of Co1.

d1=(b1b2b3b72b4b53)6,d2=(b1b2b3b73b4b52)6,d3=(b1b2b3b73b42b53)4,d4=(b1b2b32b72b42b53)6.

Consider the group generated by fixing c and a is replaced by d2 say H3 = < c, d2 >, then compute the normalizer of c inside H3. Some elements of H3, which are used in further computations, are given below:

e1=cd2,e2=cd2c,e3=cd2cd2

The words for normalizer of c inside H3 are given by:

k6=d2e1d2e15d2e115d2e15d2e113,k7=d2e15d2e16d2e16d2e16d2e15,k8=d2e114d2e114d2e114d2e114d2e114.

Now combining the normalizer of c inside H2 and H3 we get the complete normalizer of Co1 given by k5, k6 and k7. The words for (D10 × (A5 × A5).2).2 are k5k6 and k7.

3.3 Construction of 51+2GL2(5) inside Co1

From the information given in [5], the required maximal subgroup is the normalizer of an element of class 5c. The construction of this group consist of two steps given below.

Step 1

Here we first find an element of class 5c. For that we give some words of Co1. These words are given by:

b1=ababba,b2=aba,b3=abb4=abab,b5=ababa,b6=ababbb7=ababbab,b8=ababbaba,b9=ababbababb10=ababbababa,b11=ababbababab,b12=babbabababb13=babbabababa,b14=babbabababb,b15=babbabababba.

Then we use the power maps to find an element of order 5 and next check its centralizer order which confirms that the element belongs to class 5c. The element of class 5c is given by 5c = (b13b15)3.

Step 2

In this step we will find the normalizer of 5c inside Co1. The normalizer can be found by using the technique given in [10] i.e. we construct the partial normalizer of 5c inside different subgroups of of Co1. Then we combine these partial normalizer to get the required normalizer. The computations of these partial normalizers are given below.

Consider the group generated by 5c and c1 say H1 = < 5c, c1 >, then compute the normalizer of 5c inside H1. Before computing the partial normalizer we will give some words of H1 which will facilitates us in computations. These words are given by:

d1=b10e1(b10−1),d2=5cd1,d3=5cd15c,d4=5cd15c5c,d5=5cd15c5cd1.

Next we use the “TKnormalizertest” given in [11] to compute the words for the partial normalizer of 5c inside H1. These words are given by:

k1=d1d21d1d24d1d24d1d24d1d22,k2=d1d24d1d24d1d27d1d23d1d27,k3=d1d27d1d23d1d27d1d24d1d24,k4=d1d25d1d27d1d25d1d210d1d2,k5=d1d210d1d22d1d27d1d25d1d26,k80=(e1e2e32a2b2e1)6,k81=(e1e2e32a2b2e12)4.

Again consider the group generated by 5c inside k81 say H2 = < 5c, k81 >. Here we compute the partial normalizer of 5c inside H2 by giving the similar arguments as mentioned above. We will give some words for H2 which are used in computations. These words are given below:

d6=5ck81,d7=5ck815c,d8=5ck815ck81,d9=5ck815ck81k81,d10=5ck815ck81k815c,d11=5ck815ck81k815c5c,d12=5ck815ck81k815c5ck81,d13=5ck815ck81k815c5ck81k81,d14=5ck815ck81k815c5ck81k815c,d15=d6d7,d16=d6d8,d17=d6d9,d18=d6d10,d19=d6d11,d20=d6d12,d21=d6d13,d22=d6d14,c2=(k2)5.

The words for the partial normalizer are given below:

k7=c2d83c2d83c2d89c2d83c2d83,k8=c2d83c2d83c2d89c2d86c2d89,k9=c2d83c2d86c2d83c2d86c2d83.

Consider the group generated by k6, k7 and k8 say H3 = < k6, k7, k8 >. The words for H3 are given below:

d23=k6k7,d24=k6k8,d25=k7k8,d26=k6k7k8,d27=k6k7k8k6,d28=k6k7k8k7,d29=k6k7k8k7k8,d30=k6k7k8k7k8k6,d31=k6k7k8k7k8k7,d32=k6k7k8k7k8k8.

The words for the normalizer of 5c inside H3 are given below:

k9=5ck635ck635ck635ck635ck63,k10=5ck665ck665ck665ck665ck66,k11=5ck725ck725ck725ck725ck72,k12=5ck85ck825ck825ck825ck8,k13=5cd255cd2555cd2555cd2525cd256.

Consider the involution c3 = k42 . Since c3 is an involution so its normalizer can easily be calculated by using the method given by J.Bray [13]. The generators of the normalizer of c3 inside Co1 are given below:

f1=5c[(c3,5c)]2,f2=k815ck81[c3,k815ck81]16,f3=[c3,k815ck815c]12,f4=[c3,k815ck815ck81]6.

Consider the group generated by f2 and f3 say H4 = < f2, f3 >. Now compute the normalizer of 5c inside H4. Before computing the normalizer we give some words of H4. These words are given below:

d33=f2f3,d34=f2f3f2,d40=f3f2f3f3f2f2f3.

The word for the normalizer of 5c inside H4 is given below:

k14=d33d406d33d407d33d406d33d402d33d407.

Now combining the above partial normalizers will give us the words for the required maximal subgroup. These words are given by k4, k13 and k14.

3.4 Construction of 31+4:2.S4(3).2 inside Co1

From the information given in Atlas [5] the required maximal subgroup is the normalizer of an element of class 3B. So here we first find an element of class 3B and then find the normalizer of it. We will give some words of Co1. These words are given by:

b1=ababba,b2=aba,b3=ab,b4=abab,b5=ababa,b6=ababb,b7=ababbab,c1=b1b2b3b7b4b52,c2=(b1b2b3b7b4b52)3,c3=(b1b2b3b7b4b52)5,c4=(b1b2b3b7b42b53)6,c5=(b1b2b3b7b43b53)1,c6=(b1b2b3b7b43b53)4.

The construction of this group consist of two steps given below.

  1. In this step we will find an element of class 3B. This can be done by using the power maps of the above generated elements, then checking whether the centralizer order confirms that the element under consideration belongs to class 3B or not. The element of class 3B is given by “b”.

  2. In this step we will find the normalizer of b inside Co1. The normalizer can be found by using the technique given in [10] i.e. we construct the partial normalizer of b inside different subgroups of of Co1. Then we combine these partial normalizer to get the required normalizer. The computations of these partial normalizers are given below.

Consider the group generated by b and c1 say H1 = < b, c1 >, then compute the normalizer of b inside H1. Before computing the partial normalizer we will give some words of H1 which will facilitates our computations. These words are given below:

e1=bc1,e2=bc1b,e3=bc1bc1.

Next we use the “TKnormalizertest” [11] to compute the words for the partial normalizer of b inside H1.

k1=be1be1be113be111be112, k2=be1be16be110be15be112,k3=be1be113be12be18be14,k4=be12be111be16be111be12,k5=be14be19be12be111be15,k6=be14be19be13be1be16.

Again consider the group generated by b and c4 say H2 = < b, c4 >. Here we compute the partial normalizer of b inside H2 by giving similar arguments as those mentioned above. We will give some words for H2 which are used in further computations. These words are given by:

e9=bc4,e10=bc4b,e11=bc4bc4,e12=bc4bc4b.

The words for the partial normalizer are given below:

k7=be9be9be93be911be912,  k8=be9be92be92be92be92,k9=be9be913be92be99be94,k10=be9be913be94be95be96.

Now combining the above partial normalizers will give the words for the required maximal subgroup. These words are given by k5 and k10.

3.5 Construction of (A4 × G2(4)) : 2 inside Co1

From the information given in Atlas [5] the required maximal subgroup is the normalizer of 2B2 (a four group whose involutions are in class 2B). The construction of this subgroup consist of two steps.

  1. In this step we find 2B2. First we find an involution of class 2B. This involution is given by a, then we find the centralizer inside Co1. This can be done by using the technique given by J. Bray given in [13]. The generators of the centralizer inside Co1 are given by:

    a1=[a,b]3,a2=[a,ba]3,  a3=bab[a,bab]7,a4=baba[a,baba]7,  a5=babab[a,babab]5,a6=abab[a,abab]7,a7=ababa[a,ababa]7.

    Then we search inside this centralizer for an involution of class 2B to find an elementary abelian group of order 4. This involution is given by c = a515 . Now we have the required 2B2 generated by a and c where c = a515 .

  2. In this step we will calculate the normalizer of 2B2 inside Co1. The normalizer can be found by using the technique given in [10] i.e. we construct the partial normalizer of 2B2 inside different subgroups of Co1. Then we combine these partial normalizer to get the required normalizer. We also give some elements of Co1 which are used in computations:

      b1=ab,  b2=aba,b3=abab,b4=ababa  b5=ababb,  b6=ababba,b7=babba,b8=babbab,d12=(b1b2b3b7b42b53)6,d13=(b1b2b3b7b43b52)4.

Consider the group generated by a, c and d12 say H1 = < a, c, d12 >, then compute the normalizer of 2B2 inside H1. Before computing the partial normalizer we will some words of H1 which will facilitates our computations. These words are given below:

e1=ad12,e2=cd12,e3=cd12a,e4=cd12ac.

Next we use the “TKnormalizertest” [11] to compute the words for the partial normalizer of 2B2 inside Co1. These words are given below:

k1=ae1ae1ae1ae1ae112,k2=ae1ae1ae1ae12ae1,k3=ae2ae2ae211ae29ae213k4=ae2ae23ae26ae29ae22,k5=ae2ae23ae28ae23ae2

Again consider the group generated by a, c and d13 say H2 = < a, c, d13 >. Here we compute the partial normalizer of 2B2 inside H2 by giving similar arguments to those mentioned above. We give some words for H2 which are used in further computations:

e13=ad13,e14=cd13,e15=ad13c,e16=ad13ca,

The words for the partial normalizer are given by:

k9=ae13ae13ae13ae13ae139,k10=ae13ae13ae13ae136ae137.

Now combining the above partial normalizers gives us the words for the required maximal subgroup. These words are given by k4, k5 and k10.

3.6 Construction of (22+12)A8 × S3 inside Co1

From the information given in Atlas [5] the required maximal subgroup is the normalizer of 2A2 (a four group whose involutions are in class 2A). The construction of this subgroup consist of two steps.

  1. In this step we find 2A2. First we find an involution of class 2A. This involution is given by c = (ab)20, then we find the centralizer of c inside Co1. This can be done by using the technique in[13]. The generators of the centralizer of c inside Co1 are given below:

    a2=a[a1,a]2,a3=b[a1,b]2,a4=aba[a1,aba]2,a5=bab[a1,bab]2,a6=baba[a1,baba],a7=babab[a1,babab]2.

    Then searching inside this centralizer for an involution of class 2A, which combines with c, gives an elementary abelian group of order 4. This involution is given by d = (a2a6)4. Now we have the required 2A2 generated by c and d.

  2. In this step we will calculate the normalizer of 2A2 inside Co1. The normalizer can be found by using the technique given in [10] i.e. we construct the partial normalizer of 2A2 inside different subgroups of of Co1. Then we combine these partial normalizer to get the required normalizer. We also give some elements of Co1 which are used in computations:

    b1=ab,  b2=aba,b3=abab,b4=ababa,b5=ababb,  b6=ababba,b7=babba,b8=babbab,d1=(b1b2b3b7b4b52)1, d2=(b1b2b3b7b4b52)3,d3=(b1b2b3b7b4b52)5,d4=(b1b2b3b7b42b53)6,d12=(b1b2b3b73b43b5)5.

Consider the group generated by c, d and d1 say H1 = < c, d, d1 >, then compute the normalizer of 2A2 inside H1. Before computing the partial normalizer we give some words of H1 which will facilitate our computations:

e1=cd1,e2=cd1d,e3=cd1dc,e4=cd1dcdd1.

Next we use the “TKnormalizertest” [11] to compute the words for the partial normalizer of 2B2 inside Co1. These words are given by:

k1=ce1ce1ce1ce12ce1,k2=de1de12de12de15de14,k3=de1de12de14de12de13,k4=de12de1de14de1de12,k5=de12de1de14de13de16,k6=de12de1de14de15de12,k7=de2de22de22de25de24,k8=de22de2de24de2de22.

Again consider the group generated by c, d and d4 say H2 = < c, d, d4 >. Here we compute the partial normalizer of 2A2 inside H2 by giving similar arguments to those mentioned above. We give some words for H2 which are used in further computations:

e11=cd4,e12=dd4,e13=dd4c,e14=dd4cd.

The words for the partial normalizer are given by:

  k9=de11de11de112de11de11,k10=de11de11de112de11de116,k11=de11de11de115de114de114,k12=de11de112de112de11de115.

Now consider the group generated by c, d and d12 say H3 = < c, d, d12 >. Here we compute the partial normalizer of 2A2 inside H3 by by giving similar arguments to those mentioned above. We give some words for H3 which are used in further computations:

e22=cd12,e23=dd12,e24=dd12c,e25=dd12cd.

The words for the partial normalizer are given below:

k15=ce2214ce2220ce2220ce2220ce226,k16=ce237ce235ce236ce237ce235,k17=de23de234de234de238de232.

Now combining the above partial normalizers gives the words for the required maximal subgroup. These words are given by k6, k16 and k17.

3.7 Construction of 36 : 2.M12 inside Co1

From the information given in Atlas [5] the required maximal subgroup is the normalizer of 36 (elementary abelian group of order 729). The construction of this subgroup consist of two steps given below.

  1. Here we will construct 36. To construct 36 we adopt the following strategy.

    1. Find an arbitrary element of order 3. This element is given by b.

    2. Find centralizer of b inside Co1. Before calculating the centralizer we give some elements of Co1 as follows:

      b1=ab,b2=aba,b3=abab,b4=ababa,b5=ababb,b6=ababba,b7=babba,b8=babbab,c1=(b1b2b3b7b4b52),c2=(b1b2b3b7b4b52)3,c3=(b1b2b3b7b4b52)5,c4=(b1b2b3b7b42b53)6.

    The centralizer of b can be found by using the technique given in [10] i.e. we start by constructing the partial centralizer of 36 inside different subgroups of of Co1. Next we combine these partial centralizer to get the required centralizer of b inside Co1. We also give some elements of Co1 which are used in computations. Consider the group generated by b, c1 say H1 = < c, c1 >. Compute the centralizer of b inside H1. Before computing the partial centralizer we give some random elements of H1 which are used in computations. These elements are given by:

    e1=bc1,e2=bc1b,e3=bc1bc1,e4=bc1bc1b.

    The generators of centralizer of b inside Co1 are given below:

    k1=be1be1be113be111be112,k2=be1be16be110be15be112,k3=be1be113be12be18be14,k4=be12be111be16be111be12,k5=be14be19be12be111be15,k6=be14be19be13be1be16.

    Again consider the group generated by b and c4 say H2 = < b, c4 >. Here we compute the partial normalizer of b inside H2. We give some random elements of H2 which are used in further computations:

    e9=bc4,e10=bc4b,e11=bc4bc4.

    The generators of centralizer of b inside H2 are given below:

    k7=be9be9be913be911be912, k8=be9be92be92be92be92,k9=be9be913be92be99be94,k10=be9be913be94be95be96.

    Now combining the above partial normalizers gives us the generators for 36. These generators are given by k1, k2, k3, k4, k5, k6, k7, k8, k9 and k10. Then we can easily find 36 inside the above centralizer. The generators for 36 are given by:

    f1=k1,f2=k2,f3=k32,f4=k42,f5=(k2k7)2,f6=(k2k10)4.

  2. In this step we will find the normalizer of 36 inside Co1 which is the required maximal subgroup. The words for the normalizer of 36 are given below:

    z1=b12b22b3b42b53,z2=b1b22b34b4b5,x1=f2k7,z3=f3x12f3x16f3x115f3x16f3x12,z4=f3x13f3x112f3x16f3x112f3x13,z5=(f3x14)3x13(f3x14)2,x2=f6z2f5f6z2,z6=(f5x25)5.

The words for 36 : 2.M12 are w1 = z3z4 and w2 = z5z6.

3.8 Construction of 32 .U4(2).D8 inside Co1

From the information given in Atlas [5] the required maximal subgroup is the normalizer of 32 (elementary abelian group of order 9). Similarly as in the previous cases we will construct this group into two steps given below.

  1. In this step we will find 32. This can be done by taking 36 which we constructed in section 3.7, then searching inside these 36 we can easily find the required 32 given by f4 and f5.

  2. In this step we will find the normalizer of H1 = < f4, f5 > inside Co1. The normalizer can be found by using the technique given in [10] i.e., we construct the partial normalizer of H1 inside different subgroups of of Co1. Then we combine these partial normalizer to get the required normalizer. The computations of these partial normalizers are given below.

Before computing the partial normalizer we give some words of Co1 which will facilitate our computations. These words are given below:

 l1=b6b72b34b4b53, l2=b6b73b32b43b53, l3=b6b73b33b46b53, l4=b6b74b3b42b52, l5=b6b74b34b4b5, l6=b6b75b33b45b5, l7=b6b76b34b4b5, l8=b62b7b3b46b52, l9=b62b77b3b46b52,l10=b63b72b34b4b53,l11=b63b73b32b43b53,l12=b63b73b33b46b53,l13=b63b74b3b42b52,l14=b63b74b34b4b5,l15=b63b75b33b45b5,l16=b63b76b34b41b5,l17=b64b7b3b46b52,l18=b64b77b3b46b52,g1=f5l1,g2=f6l1,g3=f6l1f5,g4=f6l1f5l1,g5=f6l1f5l1f6,g6=f5l1f5l1f6.

the words for the words for the partial normalizer of H1 are:

k11=f5g42f5g45f5g44f5g45f5g42,k12=f5g47f5g46f5g45f5g46f5g47,k13=f5g1f5g1f5g1f5g1f5g14.

We find an involution inside the above calculated normalizer and then calculate its centralizer. This involution is given by g15 = k153 , generators of the centralizer of g15 inside Co1 are:

h1=[g15,a]2,h2=b[g15,b],h3=[g15,ab]3h4=[g15,aba]3,h5=[g15,abab]2.

Now we will find the partial normalizer of H1 inside Co1. The words for the centralizer are given below:

g16=h2h3,g17=h2h4,g18=h2h5,g19=h1h4h5,g20=h1h4h5h2,g21=h1h4h5h2h3,g22=h1h4h5h2h3h4,g23=h1h4h5h2h3h4h5,g24=h1h4h5h2h3h4h5h2,g25=h1h4h5h2h3h4h5h3,g26=h1h4h5h2h3h4h5h4,g27=h1h4h5h2h3h4h5h4h2,g28=h1h2h3h4h5.

The words for the partial normalizer of H1 inside Co1 are given by:

k17=h1g16h1g162h1g162h1g1611h1g168,k18=h1g16h1g162h1g168h1g1611h1g168,k19=h1g16h1g164h1g1610h1g16h1g168,k20=h2g27h2g274h2g273h2g275h2g273,k31=f5g282f5g287f5g284f5g286f5g282.

The words for the required maximal subgroup 32. U4(2).D8 are k11 and k19k31.

3.9 Construction of 33+4 : 2.(S4 × S4) inside Co1

Following [5], we see that the required subgroup is the normalizer of 33. We can easily find 33 from 36 calculated above in 3.7, then the normalizer of it gives us the required subgroup. The generators of 33 are f4, f5 and f6. Before computing the normalizer we give some elements:

 l1=b6b72b34b4b53,g1=f4l1,g7=f4l1f5l1f6, g8=f4f5f6l1f4l1,z13=f4g8f4g85f4g8f4g86f4g8,g9=z133, h1=[g9,a]3,h2=b[g9,b],h3=ab[g9,ab]2, h4=[g9,aba]3,h5=abab[g9,abab]2,g10=h2h5,g11=h1h2h3,g12=h1h3h4,g13=g11g12.

The generators for the normalizer of 33 are given below:

z11=f6g13(f6g17)3f6g14,z12=f6g7(f6g73)2f6g72f6g78,z14=h1g104(h1g103)2h1g104h1g106,z15=h1g13h1g132h1g1313h1g13h1g138,z16=(z14z15)4z152z14z157,z17=(z14z15)2z14z157z14z155z14z152.

The words for 33+4 : 2.(S4 × S4) are given by w5 = z16z17 and w6 = z12.

3.10 Construction of 24+12.(S3 × 3S6) inside Co1

From [5], the required subgroup is the normalizer of 2A4 (inside Co1) and can be constructed by taking an involution of class 2A, then searching inside its centralizer. We have already constructed NCo1(2A2) in section 3.6, and now we will search 2A4 inside NCo1(2A2). Now 2A4 = 〈k1, k3, g1, g2〉, where k1, k3 are same as in section 3.6 and g1 = (k2k8)2, g2 = (k2k9)2. The words for k8 and k9 are given in section 3.6.

h1=k1,h2=k3,h5=a[h1,a]2,h7=ba[h1,ba]2,h8=[h2,ab]2,h9=[h2,ba]3,h10=aba[h2,aba]2,l1=h8h9,l2=h8h10.

The generator for the normalizer of 2A4 are given below.

k10=h5h7h5h72h5h75h5h7h5h74,k11=h5h7h5h72h5h75h5h73h5h76,k12=l1l2l1l26l1l24l1l2l1l26.

The words for 24+12.(S3 × 3S6) are found to be k11 and k13 = k12k10.

3.11 Construction of 53 : (4 × A5).2 inside Co1

From [5], 53 : (4 × A5).2 is the normalizer of 53 inside Co1. We give some random elements of Co1 below.

b1=(ab)2ba,b3=ab,b6=(ab)2b,b10=(ab)2(ba)3b,b13=bab(ba)4,b15=babb(ab)3ba, 5c=(b13b15)3,e1=(ab)20,e2=(ab)8,e3=b63,c1=(ababa2b3)4,d1=b10e1(b10−1),d2=5cd1,k2=(d1d24)3(d1d23)2d1d27,k4=d1d25d1d27d1d25d1d210d1d2,k10=(e1e2e32a2b2e12)4,d6=5ck10,d8=5ck105ck10,c2=(k2)5,k6=c2d67c2d65c2d66c2d65c2d67,k7=c2d83c2d83c2d89c2d83c2d83,k8=c2d83c2d83c2d89c2d86c2d89,k9=5ck635ck635ck635ck635ck63,k12=5ck85ck825ck825ck825ck8,c3=k42,f2=(k105ck10)[c3,k105ck10]16,f3=[c3,k105ck105c]12,d33=f2f3,d40=f3f2f3f3f2f2f3.
k14=d33d406d33d407d33d406d33d402d33d407,g0=k22,g1=k2k12,g4=k9k12,g5=k12k14, k16=(g0g1)3g0g12,k22=(g0g4)4g0g42,k23=(g0g5)4g0g52.

The generators of 53 are h1 = k16,h2 = k22 and h3 = k23.

h4=k42,h5=[h4,a]7,h6=[h4,b]15,h9=[h4,ba]15,e7=h5h9,z1=h6e73h6e72h6e7h6e76h6e73,z2=h6e73h6e76h6e79h6e74h6e72,e13=(z1)6,e14=[e13,a]7,e15=[e13,b]15,e16=[e13,abb]15,e22=e14e15e16e15,e24=e14e15e16e15e16e14,z3=e15e229e15e222e15e224e15e227e15e225,z4=e24e226e24e226e24e228e24e22e24e224.

The generators of the normalizer of 53 are h1, z1, z3 and z4. The words for 53 : (4 × A5).2 are z5 = h1z1 and z6 = z3z4z3.

3.12 Construction of 72 : (3 × 2.S4) inside Co1

Following [5], the required subgroup is the normalizer of 72. It is constructed by taking an element of class 7B and by searching inside its centralizer we find 7B2. Before computations we give some random elements of Co1:

b1=(ab)2ba,b2=aba,b3=ab,b4=(ab)2,b5=(ab)2a,b6=(ab)2b,b7=(ab)2bab,b8=(ab)2(ba)2,b9=(ab)2(ba)2b,b10=bab(ba)3b2a.

The elements of 7B are given by a2 = b106 . The generators of 7B2 are given below:

f1=a2e1a2e1a2e13a2e17a2e17,f2=a2e1a2e12a2e18a2e1a2e16.

Now we find the normalizer of 7B2.

k1=a2e1a2e12a2e17a2e17a2e110,k2=a2e1a2e12a2e18a2e1a2e16,g2=(b7b8b96b44b62)8,e3=a2g2,e2=a2g2a2g22a22,k3=(e2e3e2e32)2e39e2e38.

The words for 72 : (3 × 2.S4) are k3 and k4 = k1k2.

3.13 Construction of 52 : 2A5 inside Co1

The required subgroup is the normalizer of 5C2 [5] and it can be constructed by taking an element of order 5 and then search inside its centralizer. We can easily find 5C2.

b1=ababb,b2=ababbababa,b3=babbabababa,b4=babbabababba,5c=(b3b4)3,e1=(ab)20,e2=(ab)8,e3=b63,d1=b2e1(b2−1),d2=5cd1,k1=d1d2d1d24d1d24d1d24d1d22,k2=d1d24d1d24d1d27d1d23d1d27,k4=d1d25d1d27d1d25d1d210d1d2,k4=(e1e2e32a2b2e12)4,d3=5ck4,d4=5ck45ck4,c2=k25,k5=c2d67c2d65c2d66c2d65c2d67,k6=c2d43c2d43c2d49c2d43c2d43,k7=c2d43c2d43c2d49c2d46c2d49,k8=5ck535ck535ck535ck535ck53,d5=k6k7,
k9=5cd55cd555cd555cd525cd56,c3=k32,f2=k45ck4[c3,k45ck4]16,f3=[c3,k45ck45c]12,d6=f2f3,d7=f3f2f32f22f3,k10=d6d76d6d77d6d76d6d72d6d77,z1=k2k10.

The generators of 5C2 are given by 5c and z1.

g1=k2k8k10,k11=(z1g1)5g1,h1=k2k9,k12=(k9h1)4h15k9h119,j1=k122,j2=a[j1,a]6,j3=b[j1,b]7,j4=[j1,ab]15,l1=j2j3j4,k13=j2l15j2l12j2l112j2l17j2l116.

The words for 52 : 2A5 are k11 and k13.

3.14 Construction of (A7 × L2(7)) : 2 inside Co1

The whole strategy for locating (A7 × L2(7)) : 2 is given in [3]. First we take an A5 of the type (2B, 3A, 5A) which is in the unique class of Co1 with normalizer N(A5) = (A5 × J2).2 [5], then find A4 inside A5 . Then we find an element of class 3A which commutes with A4 but not with A5. This 3A element with A5 extends A5 to our required A7. Finally we found that N(A5) = (A7 × L2(7)) : 2. We give some random elements of Co1 to facilitate computations.

b1=(ab)2ba,b2=aba,b3=ab,b4=(ab)2,b5=(ab)2a,b6=(ab)2b,b7=(ab)2bab,b8=(ab)2(ba)2,b9=(ab)2(ba)2b,b10=(ab)2(ba)3,b13=bab(ba)4,b15=bab2(ab)3ba,2b=b1,3a=b1514,

The generators for A5 are:

c1=(2b)b3−12, c2=(3a)b13−4, c3=(2b)b3−12(3a)b13−4.

The generators of A4 are given by d1 = c1, and d2=(c2c3)2c3c2c32c1c2c3c2.

e1=[d1,ab]3,e2=[d1,ba]6,e3=[d1,baba]6,e6=e2e3,e7=e1e2e3,f1=e1e6e1e6e1e6e1e6e1e62.

Here f5 = f17 commutes with A4 but not with A5, so the generators of A7 are c1, c2, c3 and f5.

e8=e1e6,e9=e1e7,k1=e1e8e1e85e1e84e1e811e1e86,k4=e2e85e2e82e2e89e2e82e2e87,g1=(c3f5c2c1c3)2,g2=[g1,b]10,g3=ab[g1,ab]5,g5=g2g3,k6=g2g55g2g510g2g55g2g53g2g511.

The words for (A7 × L2(7)) : 2 are k1 and k7 = k4k6.

3.15 Construction of (A6 × U3(3)) : 2 inside Co1

The required group is the normalizer of A6 which lies in the suzuki chain[3]. We easily find A6 inside 3.14. The generators of A6 are given by g1 = c1 and g2 = (f5c3)2c1c2(c3c2)2f5c3f5c1c2c3c2. Next we find the normalizer of A6.

h1=(b7b8b911b46b62)7,e8=g2h1,k2=g1e82g1e8g1e83g1e8g1e8,k4=g1e82g1e82g1e83g1e82g1e8,l6=ababa[g1,ababa]5,l8=l4l6,k7=l4l83l4l86l4l8l4l82l4l86.

The words for (A6 × U3(3)) : 2 are k8 = k7k4 and k9 = k2k4k7.

3.16 Construction of (A5 × J2) : 2 inside Co1

The required group is the normalizer of A5 [5] which we already constructed in 3.14. It can also be constructed by taking an involution of class 2B, an element of class 3A and product of these two elements belongs to class 5A [5], then N(2B, 3A, 5A) = (A5 × J2).2.

b1=(ab)2a,b2=bab2(ab)3,b3=bab2(ab)3ba,3a=b314,2b=a.

The generators of the required group whose normalizer is to be computed are given below:

c1=(3a)b1−18, c2=(2b)b2−1, 3=(3a)b1−18(2b)b2−1.

Before computing the normalizer we give some elements:

d1=a[c2,a],d2=[c2,b]12,d3=aba[c2,aba]7,d4=d1d3,k1=d2d4d2d4d2d44d2d46d2d43,e1=k112,k2=(d2d4)3d46(d2d46)2,e3=[e1,b]2,e4=[e1,ab]3,e5=aba[e1,aba]2,e6=e4e5,k3=e3e62e3e63e3e64e3e67e3e66.

The words for (A5 × J2) : 2 are k2 and k3.

The orders and orbit-shapes of above computed maximal subgroups are mentioned in table given below.

Group Order Orbit-shape
(A9 × S3) 1088640 3601 32402 201601 259201 453601
(D10 × (A5 × A5).2).2 144000 601 6001 7201 15001 30003
120001 144001 180002 240001
51+2 GL2(5) 60000 51 1501 12501 15001 18751 25002
30002 75001 150003 300001
31+4 : 2.S4(3).2 25194240 271 32401 97201 328051 524881
(22+12)A8 × S3 1981808640 3601 57601 921601
36 : 2.M12 138568320 5941 174961 320761 481141
32.U4(3).D8 235146240 7561 362881 612361
33+4 : 2. (S4 × S4) 2519424 1081 19441 87481 349921 524881
24+12.(S3 × 3S6) 849346560 721 14401 230401 737281
53 : (4 × A5).2 60000 301 1501 6001 7506 15004 300013 60008
72 : (3 × 2.S4) 3528 841 5883 8822 11767 17645 352822
52 : 4A5 3000 301 1501 3001 6003 7504 15002 300030
(A7 × L2(7)) : 2 846720 25201 352801 604801
(A6 × U3(3)) : 2 4353560 75601 907201
(A4 × G2(4)) : 2 6038323200 982801
(A5 × J2) : 2 72576000 378001 604801

Acknowledgement

The authors would like to thank anonymous referees for their valuable comments.

  1. Competing Interest: The authors do not have any competing interests.

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Received: 2018-09-18
Accepted: 2019-02-07
Published Online: 2019-04-29

© 2019 Yasin et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 Public License.

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  51. A note on the structure of a finite group G having a subgroup H maximal in 〈H, Hg〉
  52. A fuzzy multi-objective linear programming with interval-typed triangular fuzzy numbers
  53. Variational-like inequalities for n-dimensional fuzzy-vector-valued functions and fuzzy optimization
  54. Stability property of the prey free equilibrium point
  55. Rayleigh-Ritz Majorization Error Bounds for the Linear Response Eigenvalue Problem
  56. Hyper-Wiener indices of polyphenyl chains and polyphenyl spiders
  57. Razumikhin-type theorem on time-changed stochastic functional differential equations with Markovian switching
  58. Fixed Points of Meromorphic Functions and Their Higher Order Differences and Shifts
  59. Properties and Inference for a New Class of Generalized Rayleigh Distributions with an Application
  60. Nonfragile observer-based guaranteed cost finite-time control of discrete-time positive impulsive switched systems
  61. Empirical likelihood confidence regions of the parameters in a partially single-index varying-coefficient model
  62. Algebraic loop structures on algebra comultiplications
  63. Two weight estimates for a class of (p, q) type sublinear operators and their commutators
  64. Dynamic of a nonautonomous two-species impulsive competitive system with infinite delays
  65. 2-closures of primitive permutation groups of holomorph type
  66. Monotonicity properties and inequalities related to generalized Grötzsch ring functions
  67. Variation inequalities related to Schrödinger operators on weighted Morrey spaces
  68. Research on cooperation strategy between government and green supply chain based on differential game
  69. Extinction of a two species competitive stage-structured system with the effect of toxic substance and harvesting
  70. *-Ricci soliton on (κ, μ)′-almost Kenmotsu manifolds
  71. Some improved bounds on two energy-like invariants of some derived graphs
  72. Pricing under dynamic risk measures
  73. Finite groups with star-free noncyclic graphs
  74. A degree approach to relationship among fuzzy convex structures, fuzzy closure systems and fuzzy Alexandrov topologies
  75. S-shaped connected component of radial positive solutions for a prescribed mean curvature problem in an annular domain
  76. On Diophantine equations involving Lucas sequences
  77. A new way to represent functions as series
  78. Stability and Hopf bifurcation periodic orbits in delay coupled Lotka-Volterra ring system
  79. Some remarks on a pair of seemingly unrelated regression models
  80. Lyapunov stable homoclinic classes for smooth vector fields
  81. Stabilizers in EQ-algebras
  82. The properties of solutions for several types of Painlevé equations concerning fixed-points, zeros and poles
  83. Spectrum perturbations of compact operators in a Banach space
  84. The non-commuting graph of a non-central hypergroup
  85. Lie symmetry analysis and conservation law for the equation arising from higher order Broer-Kaup equation
  86. Positive solutions of the discrete Dirichlet problem involving the mean curvature operator
  87. Dislocated quasi cone b-metric space over Banach algebra and contraction principles with application to functional equations
  88. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the open semi-axis
  89. Differential polynomials of L-functions with truncated shared values
  90. Exclusion sets in the S-type eigenvalue localization sets for tensors
  91. Continuous linear operators on Orlicz-Bochner spaces
  92. Non-trivial solutions for Schrödinger-Poisson systems involving critical nonlocal term and potential vanishing at infinity
  93. Characterizations of Benson proper efficiency of set-valued optimization in real linear spaces
  94. A quantitative obstruction to collapsing surfaces
  95. Dynamic behaviors of a Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species
  96. Coexistence for a kind of stochastic three-species competitive models
  97. Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1
  98. On the two-term exponential sums and character sums of polynomials
  99. F-biharmonic maps into general Riemannian manifolds
  100. Embeddings of harmonic mixed norm spaces on smoothly bounded domains in ℝn
  101. Asymptotic behavior for non-autonomous stochastic plate equation on unbounded domains
  102. Power graphs and exchange property for resolving sets
  103. On nearly Hurewicz spaces
  104. Least eigenvalue of the connected graphs whose complements are cacti
  105. Determinants of two kinds of matrices whose elements involve sine functions
  106. A characterization of translational hulls of a strongly right type B semigroup
  107. Common fixed point results for two families of multivalued A–dominated contractive mappings on closed ball with applications
  108. Lp estimates for maximal functions along surfaces of revolution on product spaces
  109. Path-induced closure operators on graphs for defining digital Jordan surfaces
  110. Irreducible modules with highest weight vectors over modular Witt and special Lie superalgebras
  111. Existence of periodic solutions with prescribed minimal period of a 2nth-order discrete system
  112. Injective hulls of many-sorted ordered algebras
  113. Random uniform exponential attractor for stochastic non-autonomous reaction-diffusion equation with multiplicative noise in ℝ3
  114. Global properties of virus dynamics with B-cell impairment
  115. The monotonicity of ratios involving arc tangent function with applications
  116. A family of Cantorvals
  117. An asymptotic property of branching-type overloaded polling networks
  118. Almost periodic solutions of a commensalism system with Michaelis-Menten type harvesting on time scales
  119. Explicit order 3/2 Runge-Kutta method for numerical solutions of stochastic differential equations by using Itô-Taylor expansion
  120. L-fuzzy ideals and L-fuzzy subalgebras of Novikov algebras
  121. L-topological-convex spaces generated by L-convex bases
  122. An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior
  123. New error bounds for linear complementarity problems of Σ-SDD matrices and SB-matrices
  124. Hankel determinant of order three for familiar subsets of analytic functions related with sine function
  125. On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5
  126. Results on existence for generalized nD Navier-Stokes equations
  127. Regular Banach space net and abstract-valued Orlicz space of range-varying type
  128. Some properties of pre-quasi operator ideal of type generalized Cesáro sequence space defined by weighted means
  129. On a new convergence in topological spaces
  130. On a fixed point theorem with application to functional equations
  131. Coupled system of a fractional order differential equations with weighted initial conditions
  132. Rough quotient in topological rough sets
  133. Split Hausdorff internal topologies on posets
  134. A preconditioned AOR iterative scheme for systems of linear equations with L-matrics
  135. New handy and accurate approximation for the Gaussian integrals with applications to science and engineering
  136. Special Issue on Graph Theory (GWGT 2019)
  137. The general position problem and strong resolving graphs
  138. Connected domination game played on Cartesian products
  139. On minimum algebraic connectivity of graphs whose complements are bicyclic
  140. A novel method to construct NSSD molecular graphs
https://doi.org/10.1515/math-2019-0024
Keywords for this article
Conway group; maximal subgroup; generators; finite group; normalizer
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Articles in the same Issue

  1. Regular Articles
  2. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator of orders less than one
  3. Centralizers of automorphisms permuting free generators
  4. Extreme points and support points of conformal mappings
  5. Arithmetical properties of double Möbius-Bernoulli numbers
  6. The product of quasi-ideal refined generalised quasi-adequate transversals
  7. Characterizations of the Solution Sets of Generalized Convex Fuzzy Optimization Problem
  8. Augmented, free and tensor generalized digroups
  9. Time-dependent attractor of wave equations with nonlinear damping and linear memory
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  11. Almost periodic solution of a discrete competitive system with delays and feedback controls
  12. On a problem of Hasse and Ramachandra
  13. Hopf bifurcation and stability in a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge
  14. A note on the formulas for the Drazin inverse of the sum of two matrices
  15. Completeness theorem for probability models with finitely many valued measure
  16. Periodic solution for ϕ-Laplacian neutral differential equation
  17. Asymptotic orbital shadowing property for diffeomorphisms
  18. Modular equations of a continued fraction of order six
  19. Solutions with concentration and cavitation to the Riemann problem for the isentropic relativistic Euler system for the extended Chaplygin gas
  20. Stability Problems and Analytical Integration for the Clebsch’s System
  21. Topological Indices of Para-line Graphs of V-Phenylenic Nanostructures
  22. On split Lie color triple systems
  23. Triangular Surface Patch Based on Bivariate Meyer-König-Zeller Operator
  24. Generators for maximal subgroups of Conway group Co1
  25. Positivity preserving operator splitting nonstandard finite difference methods for SEIR reaction diffusion model
  26. Characterizations of Convex spaces and Anti-matroids via Derived Operators
  27. On Partitions and Arf Semigroups
  28. Arithmetic properties for Andrews’ (48,6)- and (48,18)-singular overpartitions
  29. A concise proof to the spectral and nuclear norm bounds through tensor partitions
  30. A categorical approach to abstract convex spaces and interval spaces
  31. Dynamics of two-species delayed competitive stage-structured model described by differential-difference equations
  32. Parity results for broken 11-diamond partitions
  33. A new fourth power mean of two-term exponential sums
  34. The new operations on complete ideals
  35. Soft covering based rough graphs and corresponding decision making
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  37. Sufficient and necessary conditions of convergence for ρ͠ mixing random variables
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  44. Embedding of Supplementary Results in Strong EMT Valuations and Strength
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  46. A General Version of the Nullstellensatz for Arbitrary Fields
  47. A new representation of α-openness, α-continuity, α-irresoluteness, and α-compactness in L-fuzzy pretopological spaces
  48. Random Polygons and Estimations of π
  49. The optimal pebbling of spindle graphs
  50. MBJ-neutrosophic ideals of BCK/BCI-algebras
  51. A note on the structure of a finite group G having a subgroup H maximal in 〈H, Hg〉
  52. A fuzzy multi-objective linear programming with interval-typed triangular fuzzy numbers
  53. Variational-like inequalities for n-dimensional fuzzy-vector-valued functions and fuzzy optimization
  54. Stability property of the prey free equilibrium point
  55. Rayleigh-Ritz Majorization Error Bounds for the Linear Response Eigenvalue Problem
  56. Hyper-Wiener indices of polyphenyl chains and polyphenyl spiders
  57. Razumikhin-type theorem on time-changed stochastic functional differential equations with Markovian switching
  58. Fixed Points of Meromorphic Functions and Their Higher Order Differences and Shifts
  59. Properties and Inference for a New Class of Generalized Rayleigh Distributions with an Application
  60. Nonfragile observer-based guaranteed cost finite-time control of discrete-time positive impulsive switched systems
  61. Empirical likelihood confidence regions of the parameters in a partially single-index varying-coefficient model
  62. Algebraic loop structures on algebra comultiplications
  63. Two weight estimates for a class of (p, q) type sublinear operators and their commutators
  64. Dynamic of a nonautonomous two-species impulsive competitive system with infinite delays
  65. 2-closures of primitive permutation groups of holomorph type
  66. Monotonicity properties and inequalities related to generalized Grötzsch ring functions
  67. Variation inequalities related to Schrödinger operators on weighted Morrey spaces
  68. Research on cooperation strategy between government and green supply chain based on differential game
  69. Extinction of a two species competitive stage-structured system with the effect of toxic substance and harvesting
  70. *-Ricci soliton on (κ, μ)′-almost Kenmotsu manifolds
  71. Some improved bounds on two energy-like invariants of some derived graphs
  72. Pricing under dynamic risk measures
  73. Finite groups with star-free noncyclic graphs
  74. A degree approach to relationship among fuzzy convex structures, fuzzy closure systems and fuzzy Alexandrov topologies
  75. S-shaped connected component of radial positive solutions for a prescribed mean curvature problem in an annular domain
  76. On Diophantine equations involving Lucas sequences
  77. A new way to represent functions as series
  78. Stability and Hopf bifurcation periodic orbits in delay coupled Lotka-Volterra ring system
  79. Some remarks on a pair of seemingly unrelated regression models
  80. Lyapunov stable homoclinic classes for smooth vector fields
  81. Stabilizers in EQ-algebras
  82. The properties of solutions for several types of Painlevé equations concerning fixed-points, zeros and poles
  83. Spectrum perturbations of compact operators in a Banach space
  84. The non-commuting graph of a non-central hypergroup
  85. Lie symmetry analysis and conservation law for the equation arising from higher order Broer-Kaup equation
  86. Positive solutions of the discrete Dirichlet problem involving the mean curvature operator
  87. Dislocated quasi cone b-metric space over Banach algebra and contraction principles with application to functional equations
  88. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the open semi-axis
  89. Differential polynomials of L-functions with truncated shared values
  90. Exclusion sets in the S-type eigenvalue localization sets for tensors
  91. Continuous linear operators on Orlicz-Bochner spaces
  92. Non-trivial solutions for Schrödinger-Poisson systems involving critical nonlocal term and potential vanishing at infinity
  93. Characterizations of Benson proper efficiency of set-valued optimization in real linear spaces
  94. A quantitative obstruction to collapsing surfaces
  95. Dynamic behaviors of a Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species
  96. Coexistence for a kind of stochastic three-species competitive models
  97. Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1
  98. On the two-term exponential sums and character sums of polynomials
  99. F-biharmonic maps into general Riemannian manifolds
  100. Embeddings of harmonic mixed norm spaces on smoothly bounded domains in ℝn
  101. Asymptotic behavior for non-autonomous stochastic plate equation on unbounded domains
  102. Power graphs and exchange property for resolving sets
  103. On nearly Hurewicz spaces
  104. Least eigenvalue of the connected graphs whose complements are cacti
  105. Determinants of two kinds of matrices whose elements involve sine functions
  106. A characterization of translational hulls of a strongly right type B semigroup
  107. Common fixed point results for two families of multivalued A–dominated contractive mappings on closed ball with applications
  108. Lp estimates for maximal functions along surfaces of revolution on product spaces
  109. Path-induced closure operators on graphs for defining digital Jordan surfaces
  110. Irreducible modules with highest weight vectors over modular Witt and special Lie superalgebras
  111. Existence of periodic solutions with prescribed minimal period of a 2nth-order discrete system
  112. Injective hulls of many-sorted ordered algebras
  113. Random uniform exponential attractor for stochastic non-autonomous reaction-diffusion equation with multiplicative noise in ℝ3
  114. Global properties of virus dynamics with B-cell impairment
  115. The monotonicity of ratios involving arc tangent function with applications
  116. A family of Cantorvals
  117. An asymptotic property of branching-type overloaded polling networks
  118. Almost periodic solutions of a commensalism system with Michaelis-Menten type harvesting on time scales
  119. Explicit order 3/2 Runge-Kutta method for numerical solutions of stochastic differential equations by using Itô-Taylor expansion
  120. L-fuzzy ideals and L-fuzzy subalgebras of Novikov algebras
  121. L-topological-convex spaces generated by L-convex bases
  122. An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior
  123. New error bounds for linear complementarity problems of Σ-SDD matrices and SB-matrices
  124. Hankel determinant of order three for familiar subsets of analytic functions related with sine function
  125. On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5
  126. Results on existence for generalized nD Navier-Stokes equations
  127. Regular Banach space net and abstract-valued Orlicz space of range-varying type
  128. Some properties of pre-quasi operator ideal of type generalized Cesáro sequence space defined by weighted means
  129. On a new convergence in topological spaces
  130. On a fixed point theorem with application to functional equations
  131. Coupled system of a fractional order differential equations with weighted initial conditions
  132. Rough quotient in topological rough sets
  133. Split Hausdorff internal topologies on posets
  134. A preconditioned AOR iterative scheme for systems of linear equations with L-matrics
  135. New handy and accurate approximation for the Gaussian integrals with applications to science and engineering
  136. Special Issue on Graph Theory (GWGT 2019)
  137. The general position problem and strong resolving graphs
  138. Connected domination game played on Cartesian products
  139. On minimum algebraic connectivity of graphs whose complements are bicyclic
  140. A novel method to construct NSSD molecular graphs
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