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BY 4.0 license Open Access Published by De Gruyter Open Access December 31, 2019

Generators of Maximal Subgroups of Harada-Norton and some Linear Groups

  • Jia-Bao Liu , Faisal Yasin EMAIL logo , Adeel Farooq and Absar Ul Haq EMAIL logo
From the journal Open Chemistry

Abstract

Group theory, the ultimate theory for symmetry, is a powerful tool that has a direct impact on research in robotics, computer vision, computer graphics and medical image analysis. Symmetry is very important in chemistry research and group theory is the tool that is used to determine symmetry. Usually, it is not only the symmetry of molecule but also the symmetries of some local atoms, molecular orbitals, rotations and vibrations of bonds, etc. that are important. Harada-Norton group is an example of a sporadic simple group. There are 14 maximal subgroups of Harada-Norton group. Generators (also known as words) of 11 maximal subgroups are already known. The aim of this note is to give generators of the remaining 3 maximal subgroups, which is an open problem mentioned on A World-wide-web Atlas of Group Representations (http://brauer.maths.qmul.ac.uk/Atlas) [1]. In this report we compute the generators of A6 × A6.D8, 23+2+6.(3 × L3(2)) and 34 : 2.(A4 × A4).4. Moreover we also compute the generators for the Maximal subgroups of some linear groups.

1 Introduction

Group theory is important in organic chemistry in studying symmetry of molecules [2]. Usually, all the molecules are symmetric and rotations and vibrations of bonds are important [3]. For example, from the symmetries of molecular orbital wave functions one can figure out the information about the binding [4]. From the symmetries, we can explain the transition and change the bands [3, 4]. Symmetry elements and symmetric operations are important concepts in group theory and if we apply any operation on a molecule and the molecule remains unchanged we call it symmetry operation. That means, the molecule remains same after applying any symmetric operation [5, 6, 7]. When we apply symmetric operation on a molecule, the position of itams and bounds get changes but the appeareance of moelcule remains unchanged [9]. With the help of group theory and using the symmetry of molecule, we can decide physical proeprties of molecule [10]. The symmetry of a molecule provides with the information of what energy levels the orbitals will be, what the orbitals symmetries are, what transitions can occur between energy levels, even bond order to name a few can be found, all without rigorous calculations. The fact that so many important physical aspects of molucules can be derived from symmetry is a very profound statement and this is what makes group theory so powerful [11]. The study of the maximal subgroups of sporadic simple groups began in the 1960s. Chang Choi [12, 13] found all the maximal subgroups of M24. In literature, the maximal subgroups of HSand McL groups, HN and fisher groups Fi22 and Fi23 are known. The local and non-local subgroups of Fi22, Fi23 and Fi24 are given in [16, 17, 18]. In 1979, R.A. Wilson discovered the maximal subgroups of Suzuki group [19] and Rudvlis group [20]. In 1990, Steve Linton determined the maximal subgroups of Th, Fi24 and its automorphism groups. He completely discussed the maximal subgroups in [21, 22]. In 1999, R.A. Wilson constructed the maximal subgroups of B. Recently Wilson has updated the list of maximal subgroups of the Monster Group. There are still some undetermined cases. To date there are 44 maximal subgroups of Monster. Its standard generators are given in [1]. Further, the sporadic groups can be calssified into three generations. The first generation contains the Mathieu groups. The second generation contains Co1, Co2, Co3, Suz, Mcl, HS and J2. The third generation contains the remaining 8 groups: Fi22, Fi23, Fi24, Th, HN, He and B. Finally, the Monster group itself is considered to be in this generation.

The concept of standard generators for sporadic simple groups was introduced by R. A. Wilson. He started a project known as an online version of Atlas, which would provide not only representations (matrix and permutation) but also words for the maximal subgroups of simple and almost simple groups. The words for the maximal subgroups of M12.2, M22.2, HS.2, McL.2, J2.2, Suz.2, He.2, Fi22.2, HN.2 and Fi24 are dicussed by Simon in [23]. In 2001, John N. Bray worked on the maximal subgroups of sporadic simple groups of order less than 1014. He presents a complete list by providing words for the maximal subgroups of 17 sporadic simple groups which includes M11, M12, J1, M22, J2, J3, Ru, O′N, Co3, HS, McL, Suz, He, Fi22, Co2, M24, M23 and Fi22.Words for maximal subgroups of these groups are given on the world-wide-web. However, there are still some cases to be dealt with. We pursue the work initiated by R.A. Wilson of finding words for maximal subgroups of certain sporadic simple groups. Thus, the only cases on the list of the world-wide-web Atlas of finite group representations which need to be solved are HN, Fi23, Co1, B and M.

There are still some hard cases which must be solved in order to have a complete list. In this paper we provide words for the maximal subgroups of the Harada-Norton Group. Moreover, we provide words of some linear groups i.e., L2(8), L2(8) : 2, L2(13), L2(13) : 2, L2(16), L2(17), L2(17) : 2, L2(19), L2(19) : 2, L2(23), L2(29), L2(31), L3(3), L3(3) : 2, L3(5). Ideally the words should be as short as possible. We use extensively GAP [24] and MAGMA [25] for group theoratic calculations.

Our notation follows [26]. In particular, ab = b−1ab and [a, b] = a−1b−1ab.

2 Main Results

In this section we give generators for the maximal subgroups of Harada-Norton and some Linear Groups.

2.1 Harada-Norton Group

In modern algebra, more precisely in group theory, an example of a sporadic simple group is the Harada-Norton group denoted by HN having order 214.36.56.7.11.19 = 273030912000000 3×1014. There are total 26 sporadic groups and Harada-Norton group is one of them founded in 1976 by Harada and in 1975 by Norton. By observing that the Harada-Norton group has a trivial Schur multiplier and has an order 2 outer automorphism group. Let the Higman-Sims group HS, then the Harada-Norton group has involution whose centralizer is of the form 2.HS.2.

The prime 5 assumes an exceptional part in the group. For instance, it centralizes an element of order 5 in the Monster group (which is the manner by which Norton thought that it was), and thus acts normally on a vertex operator algebra over the field with 5 element [27]. This infers it follows up on a 133 dimensional algebra over F5 with a commutative however nonassociative product, practically equivalent to the Griess algebra [28].

Conway and Norton proposed in their 1979 paper [29] that monstrous moonshine isn’t constrained to the monster, yet comparative wonders might be found for different groups. Larissa Queen [30] and others in this manner found that one can develop the extensions of numerous Hauptmoduln from simple combinations of dimensions of sporadic group. For HN, the pertinent McKay-Thompson series is T5A(τ) where one can set the constant a(0) = −6,

j5Aτ=T5Aτ6=ητη5τ6+53η5τητ6=1q6+134q+760q2+3345q3+12256q4+39350q5+.s

where η(t) denotes Dedekind eta function.

The Harada-Norton group has been studied extensively in recent years and many papers are written on this group, here we mention a few [31, 32, 33, 34, 35, 36, 37, 38, 39]. Monomial modular representations and symmetric generation of the Harada-Norton group. The uniqueness of this group was proved in [32]. Ryba et al. [41] found matrix generators for this group and in [42] Norton and Wilson found all maximal subgroups of the Harada-Norton group in 1986. The following 14 are the maximal subgroups of Harada-Norton group.

1. A12

2. 2.HS.2

3. U3(8)) : 3

4. 21+8.(A5 × A5).2

5. (D10 × U3(5)).2

6. 51+4.21+4.5.4

7. 26.U4(2)

8. *A6 × A6.D8

9. *23+2+6.(3 × L3(2))

10. 52+1+2.4.A5

11. M12 : 2

12. M12 : 2

13. *34 : 2.(A4 × A4).4

14. 31+4 : 4.A5

It is an interesting problem to find the generators of a group. The Atlas of group representations contains the words for 11 maximal subgroups of HN except the 3 cases marked by asterisk. In this report we determine the generators for the above mentioned subgroups as words in the generators of HN. It is well known that if G is a simple group, M is the maximal subgroup of G and K is the minimal normal subgroup of M, then M = NG(K). The cases we have dealt with, occur as normalizers of elementary abelian groups and the required information is provided in [26]. Thus we see that N(2B3) = 23+2+6.(3 × L3(2)) and N(34) = 34 : 2.(A4 × A4).4. The normalizers were computed by the methods given in [23].We have used GAP [24] and MAGMA [25] for computations.

2.1.1 Generators of (A6 × A6) : D8

We want to work inside the subgroups as much as possible. We see that H = (A6 × A6) : 22 < A12 < HN, so all we need is to construct H inside A12 and then find an involution inside HN which extends H to (A6 × A6) : D8. The details are as follows.

It is trivial to find (A6 × A6) inside A12. Next we find an involution inside NA12 (A6 × A6), which extends (A6 × A6) to (A6 × A6) : 2. We find another involution which extends (A6 × A6) : 2 to H. Now we want to use this working inside HN.

The standard generators of A12 inside HN can be constructed by observing that the 3A and 11A classes of A12 fuse to 3A and 11A classes of HN . After obtaining the standard generators of A12, we lift A12 inside HN = 〈a, b〉, where a, b are as in [1]. As a final step, we find an involution inside NHN(22) which extends H to (A6 × A6) : D8. The computational details are given below.

First we download the standard generators of A12 from Atlas given by c, d. Then we find the Centralizer of A6 inside A12. We now give the details of computing the centralizer of A6 inside A12, for that first we consider the standard generators of A6 given in Atlas c, d next we convert the c and d in terms of standard generators of A12 which is given by

x1=cx2=d1cd2cd3c

here x1, x2 are generators of A6 inside A12. now we find the centralizer of A6 inside A12 which includes the following computations given by

b6=(cd4cd7cd)5c(cd4cd7cd)5i1=b2c3=x2e3=(i1x22i1x2)2x2i1x22i1x21i1u1=b6u9=c35e3u16=u1u9

Here u16 and u6 are generators of centralizer of A6 inside A12. Then the generators of A6 plus the generators of Centralizer of A6 inside A12 gives us A6 × A6 given by

v1=u1u9x2v2=x2u6x1

Here v1, v2 are generators of A6 × A6.

Now we find the normalizer of A6 ×A6 inside A12. This normalizer contain an involution given by

v3=(c3d11c3d9c1)15

which extend the group A6 × A6 to A6 × A6 : 2. Similarly in the same way we can find the normalizer of A6 × A6 : 2 inside A12 and this normalizer contains an involution given by

v4=(c3d10c2d3c3)15

which extends A6×A6 : 2 to A6×A6 : 22. Here all the calculations are inside A12 and A6 × A6 : 22 is the maximal subgroup of A12 and it is not possible to extend A6 × A6 : 22 to A6 × A6 : D8 so our next target is to uplift the whole structure inside HN. Before uplifting we have to calculate the standard generators of A12 inside HN. The generators of A12 inside HN are given in [1], now we use these generators to find the standard generators of A12 inside HN. The words for the generators of A12 are given by c and d. Before this we will give some random elements.

a1=(cd)2a2=cda3=a1a210a12a2a1a2a2a2a12a25a1a4=a37

with the help of a power maps search inside the 3A and 11A classes, we found the standard generators of A12 are given by.

x=a4a22
y=a12

It is easy to uplift the structure because we have the standard generators of A12. After uplifting A6 × A6 : 22 inside HN we just need one more involution which gives us the required subgroup. It is not an easy task to find the last involution inside HN by random searching. So first here we find the normalizer of A6 × A6 : 22 inside HN, then searching an involution inside this normalizer such that this involution extends A6 × A6 : 22 to (A6 × A6) : D8 and combining this involution with v3,v4 will give us D8. The words for the normalizer of A6×A6 : 22 inside HN are given below.

w8=w42bw1bw442w42b2w32w9=w4w5bw32a3w2b3w12b2w10=aba12ba2b3a3w4b22w11=w3a3b3a6aw32b22ba22w12=w7w32b2a2w64b2aw32a2w13=a2b2ab5w3w12w43w5b1w14=a1w4w32a3w2b3w12b2w3a1b33w15=a1ba2b2w6w43a4a2b42w16=a2bb2a2w42b2a3w42b32w17=a3ba6bw32bw44b2a2w18=w4w32b2w3w43b2a2b2a2w19=bw44b2ab4a3w4b22w3a3b3w20=a6aw32b22ba22w7w32w21=b2a2w64b2aw32a2a2b2w22=ab5w3w12w43w5b1w32a3w23=w2b3w12b2w3a1b2ab3a3w24=w4b22w3a3b3a6aw32b22w25=ba22w7w32b2a2w64b2w26=aw32a2a2b2ab5w3w12w27=w43w5b1y1=w32a3w2b3w12b2aba12w42
y2=bw32a3bw6a3b3w12b2w3a1bw28=w32a3w2b3w12b2w3a1y2y12w29=w32a3w4b2a3b3w12b2aba12w30=w4w32a3w2b3w12b2w3a1b2w31=a1b2a22w4b2w43a4a2b42w32=a2bb2a2w42b2a3w42b32w32=a3ba6bw32bw44b2a2ba2w43b5w34w33=bw33a1bw32a3w2b2b2a22w34=bw33a1b2ab3a3w4b22w3w35=a3b3a6aw32b22ba22w7w36=w32b2a2w64b2aw32w37=a2a2b2ab5w3w12w43w38=w52w42b3a3w4b22w3a3w39=b3a6aw32b22ba22w7w40=w32b2a2w64b2aw32a2w41=a2b2ab5w3bw6a2w43w52b2aw42=w32a3w2b3w12b2w3a1b3w43=w32a3w2b3w12b2w3a1b2a1w42w44=w32a3w2b3w12b2w3a1b2a1y3=w8w9w10w11w12w13w14w15w16w17y4=w18w19w20w21w22w23w24w25w26w27w28w29y5=w30w31w32w33w34w35w36w37w38w39w40w41y6=w42w43w44y7=y3y4y5y6

The words for (A6 ×A6) : D8 are f1, f2 and y7 and these three generators can be converted into the two generators given below.

f1y7,f2y7

We use an orbit shape to search for a conjugate of the subgroup we just found to reduce the word length of the generators. Thus we have

(A6×A6):D8=a,d,

where d = ((ab)2a2(ba)4(ab)6a((ba)3)9)5

2.1.2 Generators of 23+2+6.(3 × L3(2))

Here our required subgroup is the N HN(2B3) [26]. From[42] we know that there are two classes of 2B-pure subgroups inside HN. The first type is generated by the center and any other 2B-involutions such that these two involutions are taken from the extra special group 21+8 inside the centralizer of 2B-involution [42].

The group 21+8 can be constructed by finding the centralizer of a 2B element. Then inside this centralizer, search for elements of order 4, 8, 12, 16, 24 or 32. Then power up these elements to obtain involutions which generate 21+8. Now searching inside 21+8, one can easily find a 2B3. The details of computing the 2B3 are given below:

The element of 2B is given by a1 = ((ba)4b(ba)3b(ba)6b2ab2)4. The generators of centralizer of a1 inside HN are given by: b1 = [a1, a]

b2=a1,b10b3=aba1,ab5b4=ab2abaa1,ab2abab5=ab2ab3aa1,ab2ab3a12b6=bab2ab5bab2aba1,bab22ab5bab2ab7

Thus generators of 2B3 are b7=b610, b8 = (b1b2b4)6and b9 = (b1b3b5)6. The generators for normalizer of 2B3 inside HN are given below.

c1=(ab)2b29,(ab)27c2=b29,(ab)2a7d1=b34,a2d2=b34,b11d3=(ab)b34,ab7

The generators for 23+2+6.(3 × L3(2)) are:

k1=c6c54c52c6c59andk2=d1d33d1d332.

2.1.3 Generators of 34 : 2.(A4 × A4).4

Following [26], we see that the required subgroup is the normalizer of 34 (inside HN). It turns out that 34 we seek is in the following chain of subgroups.

34<3+1+4:4A5<HN.

The generators for 31+4 : 4.A5 which were copied from [1] are given below.

c1=(ab)9(ab)3b10(ab)9d1=(abb)8b(abb)8

Now we give some random elements of 31+4 : 4.A5.

e2=c1d12d1c1d1b1=c1e2c1e23(c1e27)2c1e211b2=c1e2c1e24c1e210c1e242b3=c1e2c1e27c1e23b4=c1e2c1e22c1e211c1e29c1e2

The generators of 34 are b1, b2, b3 and b42.Before computing the normalizer we give some random elements of 31+4 : 4.A5.

c2=c1d1c3=ab1k3=(ac3)4c35ac37c4=bb1c5=b1c4k4=(b1c52b1c56)2b1c5c6=k3k4k1=(c1c2)5k2=c1c2c1c22c1c27c1c23c1c24k5=(k3c6)4c6k3c6

The generators of the normalizer of 34 inside HN are given below.

k1=c1c25k2=c1c2c1c22c1c27c1c23c1c24k5=k3c64c6c6k3c6

Thus it turns out that 34 : 2.(A4 × A4).4 = 〈k2, k1k5. The orbit shapes and order of the above subgroups ofHN are given below.

GroupOrderOrbitshape
(A6 × A6) : D81036800217214001450172031296143202
540017200481001108003144001162001
21600232400364800486400310368012592001
23+2+6.(3 × L3(2))10321926414481672126881107523143361215043
573441645121860162129024117203222580481
34 : 2.(A4 × A4).49331261721576164821296214581388815184158324
777661036821166441555222332873110434665611933122

2.2 Linear Groups

In this section we provide words for the maximal subgroups of L2(8), L2(8) : 2, L2(13), L2(13) : 2, L2(16), L2(17), L2(17) : 2, L2(19), L2(19) : 2, L2(23), L2(29), L2(31), L3(3), L3(3) : 2, L3(5) Ideally the words should be as short as possible.

Most often, the subgroups have been generated by two elements by using random searching in [44]. This method is quite successful if one of the short words is a and is in a very small conjugacy class. One can then search by generating subgroups using those short words. The generators for the maximal subgroup 23 : 7 of L2(8), 9 : 6 and L2(8) of the group L2(8) : 2, D14 of the group L2(13), D28 of the group L2(13) : 2, D32 and D36 of the group L2(17) : 2, D20 of the group L2(19), D36 and 19 : 18 of the group L2(19) : 2, D24 of the group L2(23), A5 of the group L2(29), D30, D32 and S4 of the group L2(31), 32 : 2S4 of the group L3(3) were computed by random rearching. There are still some hard cases in which this method is not of much use.

The next method depends on the information given in Atlas of Finite Simple Groups [26]. Following Atlas, the maximal subgroup 31 : 15 of L2(31) is computed by taking the normalizer of 31AB i.e., N(31AB) = 31 : 15. Similarly N(2A) = D32 and N(3A) = D30. The normalizer here is computed by the methods given in [43] and the programmes given by simon [23] with a little change in them.

Table 1

Maximal subgroups of L2(8).

SubGroups1stgenerator2ndgenerator
D18b2ab−1(ab)2(ab−1)2aba(ba)2(b−1a)2(ba)2b−1a
D14abb(abab−1)3ab−1
23 : 7ab(abab−1)2a

Following Atlas, the subgroup D18 is the normalizer of 3A. i.e., D18 = N(3A). Similarly D14 = N(7ABC) and 23 : 7 = N(2A3).

The subgroups 9 : 6, 7 : 6 and L2(8) were computed by random rearching, while 23.7 : 3 is computed by the information given in Atlas [34], i.e., 23.7 : 3 = N(2A3).

Following Atlas, D14 = N(7ABC), D12 = N(2A), A4 = N(2A2) and 13 : 6 = N(13AB).

Table 2

Maximal subgroups of L2(8) : 2.

SubGroups1stgenerator2ndgenerator
9 : 6ab(abab−1)2a
7 : 6ba((ab)3bab2(ab)2b(ab)3b(ab)5)
l2(8)a((ab)3bab2)((ab)3bab2(ab)2b(ab)3b(ab)4)
23.7 : 3b2ab−1(ab)2(ab−1)2aba(ba)2(b−1a)2(ba)2b−1a
Table 3

Maximal subgroups of L2 (13).

SubGroups1stgenerator2ndgenerator
D14a(ab)2b
D12(bab−1a)2(ba)2b(ab−1)2a
A12(bab−1a)2(ba)2(b−1abab−1a)2
13 : 6abb(ab−1)5(ab)2a
Table 4

Maximal subgroups of L2(13) : 2.

SubGroups1stgenerator2ndgenerator
S4b(ba)3(b−1a)2b−1
D28aab−1ab(ba)2
D24b−1a(bab−1ab)2ab2abab2(ab−1)3(ab)2a
L2(13)bab2ab−1ba(bab)2ab−1(ab)2

The maximal subgroup D28 is computed by random rearching, while the remaining four were constructed by the information given in Atlas D28 = C(2B), D24 = N(3A) and S4 = N(2A2).

Table 5

Maximal subgroups of L2 (16).

SubGroups1stgenerator2ndgenerator
D30abab−1(ab)3(ab−1)3aba
D34abab−1(bab−1a)4bab−1
A5b(bab−1a)4bab−1
24 : 15ab(bab−1a)4bab−1

The maximal subgroup A5 is computed by random rearching, while the remaining maximal subgroups were constructed by the information given in Atlas D34 = N(17AH), D30 = N(3A) and 24 : 15 = N(2A4).

Table 6

Maximal subgroups of L2 (17).

SubGroups1stgenerator2ndgenerator
17 : 8ab(ba)2(b−1a)2(ba)3
D18(bab−1a)2bab−1(bab−1a)2
D16(ba)3(b−1a)3(ba)2((b−1a)2ba)2b−1ab
S4(abab−1(ab)2)2ababab−1(ab−1ab)2abab−1a

The maximal subgroups were constructed by the information given in Atlas, i.e., 17 : 8 = N(17AB), S4 = N(2A2), D18 = 3A and D16 = N(2A).

Table 7

Maximal subgroups of L2(17) : 2.

SubGroups1stgenerator2ndgenerator
D32a((ab)4b)4
D36a((ba)4b−1a)2b
L2(17)b(ababb)3
17 : 16(ab)6ab−1ab−1(b−1a)5(ba)3b−1ab−1

The maximal subgroups D32, L217 and D36 were computed by random rearching, while the subgroup 17 : 16 is constructed by the information given in Atlas i.e., 17 : 16 = N(17AB).

Table 8

Maximal subgroups of L2 (19).

SubGroups1stgenerator2ndgenerator
19 : 9abbab−1(ab)2(abab−1)2(ab−1)2
D20ab(ab−1)3(ab)3ab−1
D18abab−1(abab−1)4a
A5bb(ab−1)3(ab)3ab−1

Following Atlas, 19 : 9 = N(19AB), A5 = N(2A, 3A, 5AB), D20 = N(2A) and D18 = N(3A).

Table 9

Maximal subgroups of L2(19) : 2.

SubGroups1stgenerator2ndgenerator
L2(19)b(ab)2
D40ab(ab)3(ab(ab−1)2)2(ab)2ab−1(ab)2a
D36a(ba)7(b−1a)2b(ab−1)3aba
S4bb−1ab−1(ab−1ab)2(ab)2
19 : 18a(ab−1(ab)2)2abab

The maximal subgroups L2(19), D36, S4 and 19 : 18 were computed by random rearching, while the subgroup D40 is constructed by the information given in Atlas i.e., D40 = N(5AB).

Table 10

Maximal subgroups of L2 (23).

SubGroups1stgenerator2ndgenerator
23 : 11abbab−1(ab−1ab)2(ab−1)2(ab)2a
D24a(bab−1a)4bab−1
S4(ba)5(b−1a)2b(abab−1)2(ba)5(b−1a)4
D22((ab)2ab−1)2(b−1aba)2b(ab−1)4abab−1a

The maximal subgroups D22), D24, S4 and 23 : 11 was constructed by the information given in Atlas i.e., D22 = C(2B), D24 = N(2A), S4 = N(2A2) and 23 : 11 = N(23AB).

Table 11

Maximal subgroups of L2 (29).

SubGroups1stgenerator2ndgenerator
29 : 14ab(ba)3b(ab−1abab−1)2ab−1
D28(ba)2(b−1a)2(ba)2b−1ab−1(ba)2b−1(ab)3(ab−1)2a
D30((ab)2ab−1)2(ba)2(b−1a)2(ba)2b−1ab−1
A5ab−1ab−1(ab)7ab−1(ab)2a

The maximal subgroup A5 is computed by random rearching, while the subgroups D28, D30 and 29 : 14 were constructed by the information given in Atlas i.e., D28 = N(2A), D30 = N(3A) and 29 : 14 = N(29AB).

Table 12

Maximal subgroups of L2 (31).

SubGroups1stgenerator2ndgenerator
31 : 15abab−1ab(ab−1)4ab(ab−1)2
D30abab−1
D32ab−1(ab−1(ab)2)2(ab)4ab−1aba
A5bb−1(ab−1(ab)2)2(ab)4ab−1aba
S4ab2(ab2)2(ab)2

The maximal subgroups D30, D32 and S4 were computed by random rearching, while the subgroup A5, and 31 : 15 were constructed by the information given in Atlas i.e., A5 = N(2A, 3A, 5AB), and 31 : 15 = N(31AB).

Table 13

Maximal subgroups of L3 (3).

SubGroups1stgenerator2ndgenerator
32 : 2S4aab2(ab)2
S4abab−1((ab−1)2(ab)2)2abab−1((ab−1)2(ab)2)2(ab−1)3
13 : 3ab(ba)6b−1(ab)2ab−1aba

The maximal subgroups 32 : 2S4, S4 and 13 : 3 were constructed by the information given in Atlas i.e., 32 : 2S4 = N(3A2), S4 = N(2A2) and 13 : 3 = N(13ABCD).

Table 14

Maximal subgroups of L3(3) : 2.

SubGroups1stgenerator2ndgenerator
L3(3)(b−1aba)2((ab−1)2(ab)2)2(ab−1)3
S4 : 2(b2a)4(b−1a)2b−1(ab2)2(ab)3b2(ab−1)5abab−1(ab)2
2.S4.2((ab)2a)2b(ab)3(ba)2b((ba)2b2)6((ba)2b2a)9
31+2.D8a(bab)2ab(ba)3(b−1a)2b−1(ab2)2a
13 : 6(ab)2a(ab)4(ab)3(ba)2b((ba)2b2)2a

The maximal subgroups L3(3), 2.S4.2, 31+2.D8 and 13 : 6 were constructed by random searching, while the subgroup S4 : 2 is constructed by the information given in Atlas i.e., S4 : 2 = C(2B).

Table 15

Maximal subgroups of L3(5).

SubGroups1stgenerator2ndgenerator
52 : GL2(5)bab(ba)3b−2ab−1aba−1b−1a−1(bab)2(ab−1)2b−1a
S5(ab)2a(ab)5(ab)3(ba)2b(ba)45((ba)3b)3
42 : S3(ab)2a(ab)5(ab)3((ba)2b)3(ba)21((ba)3b)7
S5(ab)2a(ab)5(ab)3((ba)2b)5(ba)48((ba)3b)8

The maximal subgroups 52 : GL2(5), S5, and 42 : S3 were constructed by the information given in Atlas i.e., 52 : GL2(5) = N(5A2), S5 = N(2A, 3A, 5AB), and 42 : S3 = N(2A2).

3 Conclusions

Mathematical tools help to solve many problems arising in chemistry and other areas of sciences [45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60], for example, graph theory help us to know about structural and physico-chemcial properties of chemcial compounds without using wet labs [50, 51, 52, 53, 54]. The finite groups are helpful in studying symmetry of molcules because almost all organic and inorganic compounds are symmetric about its center [56, 57, 58, 59, 60]. Our aim is to study some finite groups of higher order and find their words. In this paper we provide generators for the maximal subgroups of Harada-Norton and some linear groups. In the world-wide-web Atlas of Group Representations [1], there is only one copy of S5 in the list of maximal subgroups of L3(5), but here we provide generators for two non-conjugate copies of S5.

Acknowledgements

We are thakful to the reviewers for positive suggestions that improve the quality of this paper. The third author would like to thank Prof. R. A. Wilson for teaching him everything written in this paper while he was his student.

  1. Data Availability Statement: All data required for this research is available in this paper.

  2. Ethical approval: The conducted research is not related to either human or animal use.

  3. Author Contribution: All authors contribute equally in this paper.

  4. Funding Statement: The work was supported in part by Funding: This research was funded by the natural science research key project from Education Department of Anhui Province (Grant No. KJ2017A492), youth research special fund project of Anhui Jianzhu University (Grant No.2011183-8).

  5. Completing Interest: The authors do not have any competing interests.

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Received: 2018-09-26
Accepted: 2018-11-20
Published Online: 2019-12-31

© 2019 J.-B. Liu et al., published by De Gruyter

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

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