## 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 *A*_{6} × *A*_{6}.*D*_{8}, 2^{3+2+6}.(3 × *L*_{3}(2)) and 3^{4} : 2.(*A*_{4} × *A*_{4}).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 *M*_{24}. In literature, the maximal subgroups of *HS*and *McL* groups, *HN* and fisher groups *Fi*_{22} and *Fi*_{23} are known. The local and non-local subgroups of *Fi*_{22}, *Fi*_{23} and *Fi*_{24} 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*, *Fi*_{24} 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 *Co*_{1}, *Co*_{2}, *Co*_{3}, *Suz*, *Mcl*, *HS* and *J*_{2}. The third generation contains the remaining 8 groups: *Fi*_{22}, *Fi*_{23}, *Fi*_{24}, *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 *M*_{12}.2, *M*_{22}.2, *HS*.2, *McL*.2, *J*_{2}.2, *Suz*.2, *He*.2, *Fi*_{22}.2, *HN*.2 and *Fi*_{24} 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 *M*_{11}, *M*_{12}, *J*_{1}, *M*_{22}, *J*_{2}, *J*_{3}, *Ru*, *O′N*, *Co*_{3}, *HS*, *McL*, *Suz*, *He*, *Fi*_{22}, *Co*_{2}, *M*_{24}, *M*_{23} and *Fi*_{22}.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*, *Fi*_{23}, *Co*_{1}, *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*., *L*_{2}(8), *L*_{2}(8) : 2, *L*_{2}(13), *L*_{2}(13) : 2, *L*_{2}(16), *L*_{2}(17), *L*_{2}(17) : 2, *L*_{2}(19), *L*_{2}(19) : 2, *L*_{2}(23), *L*_{2}(29), *L*_{2}(31), *L*_{3}(3), *L*_{3}(3) : 2, *L*_{3}(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, *a ^{b}* =

*b*

^{−1}

*ab*and [

*a*,

*b*] =

*a*

^{−1}

*b*

^{−1}

*ab*.

## 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 2^{14}.3^{6}.5^{6}.7.11.19 = 273030912000000 ^{14}. 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 *F*_{5} 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 *T*_{5A}(*τ*) where one can set the constant *a*(0) = −6,

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. *A*_{12}

2. 2.*HS*.2

3. *U*_{3}(8)) : 3

4. 2^{1+8}.(*A*_{5} × *A*_{5}).2

5. (*D*_{10} × *U*_{3}(5)).2

6. 5^{1+4}.2^{1+4}.5.4

7. 26.*U*_{4}(2)

8. **A*_{6} × *A*_{6}.*D*_{8}

9. *2^{3+2+6}.(3 × *L*_{3}(2))

10. 5^{2+1+2}.4.*A*_{5}

11. *M*_{12} : 2

12. *M*_{12} : 2

13. *3^{4} : 2.(*A*_{4} × *A*_{4}).4

14. 3^{1+4} : 4.*A*_{5}

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* = *N _{G}*(

*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*(2

*B*

^{3}) = 2

^{3+2+6}.(3 ×

*L*

_{3}(2)) and

*N*(3

^{4}) = 3

^{4}: 2.(

*A*

_{4}×

*A*

_{4}).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 (*A*_{6} × *A*_{6}) : *D*_{8}

We want to work inside the subgroups as much as possible. We see that *H* = (*A*_{6} × *A*_{6}) : 2^{2} < *A*_{12} < *HN*, so all we need is to construct *H* inside *A*_{12} and then find an involution inside *HN* which extends *H* to (*A*_{6} × *A*_{6}) : *D*_{8}. The details are as follows.

It is trivial to find (*A*_{6} × *A*_{6}) inside *A*_{12}. Next we find an involution inside *N*_{A12} (*A*_{6} × *A*_{6}), which extends (*A*_{6} × *A*_{6}) to (*A*_{6} × *A*_{6}) : 2. We find another involution which extends (*A*_{6} × *A*_{6}) : 2 to *H*. Now we want to use this working inside *HN*.

The standard generators of *A*_{12} inside *HN* can be constructed by observing that the 3*A* and 11*A* classes of *A*_{12} fuse to 3*A* and 11*A* classes of *HN* . After obtaining the standard generators of *A*_{12}, we lift *A*_{12} inside *HN* = *〈a*, *b〉*, where *a*, *b* are as in [1]. As a final step, we find an involution inside *N _{HN}*(22) which extends

*H*to (

*A*

_{6}×

*A*

_{6}) :

*D*

_{8}. The computational details are given below.

First we download the standard generators of *A*_{12} from Atlas given by *c*, *d*. Then we find the Centralizer of *A*_{6} inside *A*_{12}. We now give the details of computing the centralizer of *A*_{6} inside *A*_{12}, for that first we consider the standard generators of *A*_{6} given in Atlas *c*, *d* next we convert the *c* and *d* in terms of standard generators of *A*_{12} which is given by

here *x*_{1}, *x*_{2} are generators of *A*_{6} inside *A*_{12}. now we find the centralizer of *A*_{6} inside *A*_{12} which includes the following computations given by

Here *u*_{16} and *u*_{6} are generators of centralizer of *A*_{6} inside *A*_{12}. Then the generators of *A*_{6} plus the generators of Centralizer of *A*_{6} inside *A*_{12} gives us *A*_{6} × *A*_{6} given by

Here *v*_{1}, *v*_{2} are generators of *A*_{6} × *A*_{6}.

Now we find the normalizer of *A*_{6} ×*A*_{6} inside *A*_{12}. This normalizer contain an involution given by

which extend the group *A*_{6} × *A*_{6} to *A*_{6} × *A*_{6} : 2. Similarly in the same way we can find the normalizer of *A*_{6} × *A*_{6} : 2 inside *A*_{12} and this normalizer contains an involution given by

which extends *A*_{6}×*A*_{6} : 2 to *A*_{6}×*A*_{6} : 2^{2}. Here all the calculations are inside *A*_{12} and *A*_{6} × *A*_{6} : 2^{2} is the maximal subgroup of *A*_{12} and it is not possible to extend *A*_{6} × *A*_{6} : 2^{2} to *A*_{6} × *A*_{6} : *D*_{8} so our next target is to uplift the whole structure inside *HN*. Before uplifting we have to calculate the standard generators of *A*_{12} inside *HN*. The generators of *A*_{12} inside *HN* are given in [1], now we use these generators to find the standard generators of *A*_{12} inside *HN*. The words for the generators of *A*_{12} are given by *c* and *d*. Before this we will give some random elements.

with the help of a power maps search inside the 3A and 11A classes, we found the standard generators of *A*_{12} are given by.

It is easy to uplift the structure because we have the standard generators of *A*_{12}. After uplifting *A*_{6} × *A*_{6} : 2^{2} 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 *A*_{6} × *A*_{6} : 2^{2} inside *HN*, then searching an involution inside this normalizer such that this involution extends *A*_{6} × *A*_{6} : 2^{2} to (*A*_{6} × *A*_{6}) : *D*_{8} and combining this involution with *v*_{3},*v*_{4} will give us *D*_{8}. The words for the normalizer of *A*_{6}×*A*_{6} : 2^{2} inside *HN* are given below.

The words for (*A*_{6} ×*A*_{6}) : *D*_{8} are *f*_{1}, *f*_{2} and *y*_{7} and these three generators can be converted into the two generators given below.

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

where *d* = *((ab) ^{2}a^{2}(ba)^{4}(ab)^{6}a((ba)^{3})^{9})^{5}*

#### 2.1.2 Generators of 2^{3+2+6}.(3 × *L*_{3}(2))

Here our required subgroup is the *N _{HN}*(2

*B*

^{3}) [26]. From[42] we know that there are two classes of 2

*B*-pure subgroups inside

*HN*. The first type is generated by the center and any other 2

*B*-involutions such that these two involutions are taken from the extra special group 2

^{1+8}inside the centralizer of 2

*B*-involution [42].

The group 2^{1+8} can be constructed by finding the centralizer of a 2*B* 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 2^{1+8}. Now searching inside 2^{1+8}, one can easily find a 2*B*^{3}. The details of computing the 2*B*^{3} are given below:

The element of 2*B* is given by *a*_{1} = ((*ba*)^{4}*b*(*ba*)^{3}*b*(*ba*)^{6}*b*^{2}*ab*^{2})^{4}. The generators of centralizer of *a*_{1} inside *HN* are given by: *b*_{1} = [*a*_{1}, *a*]

Thus generators of 2*B*^{3} are *b*_{8} = (*b*_{1}*b*_{2}*b*_{4})^{6}and *b*_{9} = (*b*_{1}*b*_{3}*b*_{5})^{6}. The generators for normalizer of 2*B*^{3} inside *HN* are given below.

The generators for 2^{3+2+6}.(3 × *L*_{3}(2)) are:

#### 2.1.3 Generators of 3^{4} : 2.(*A*_{4} × *A*_{4}).4

Following [26], we see that the required subgroup is the normalizer of 3^{4} (inside *HN*). It turns out that 3^{4} we seek is in the following chain of subgroups.

The generators for 3^{1+4} : 4.*A*_{5} which were copied from [1] are given below.

Now we give some random elements of 3^{1+4} : 4.*A*_{5}.

The generators of 3^{4} are *b*_{1}, *b*_{2}, *b*_{3} and ^{1+4} : 4.*A*_{5}.

The generators of the normalizer of 3^{4} inside *HN* are given below.

Thus it turns out that 3^{4} : 2.(*A*_{4} × *A*_{4}).4 = *〈k*_{2}, *k*_{1}*k*_{5}*〉*. The orbit shapes and order of the above subgroups of*HN* are given below.

Group | Order | Orbitshape |
---|---|---|

(A_{6} × A_{6}) : D_{8} | 1036800 | 2^{1}72^{1}400^{1}450^{1}720^{3}1296^{1}4320^{2} |

5400^{1}7200^{4}8100^{1}10800^{3}14400^{1}16200^{1} | ||

21600^{2}32400^{3}64800^{4}86400^{3}103680^{1}259200^{1} | ||

2^{3+2+6}.(3 × L_{3}(2)) | 1032192 | 64^{1}448^{1}672^{1}2688^{1}10752^{3}14336^{1}21504^{3} |

57344^{1}64512^{1}86016^{2}129024^{1}172032^{2}258048^{1} | ||

3^{4} : 2.(A_{4} × A_{4}).4 | 93312 | 6^{1}72^{1}576^{1}648^{2}1296^{2}1458^{1}3888^{1}5184^{1}5832^{4} |

7776^{6}10368^{2}11664^{4}15552^{2}23328^{7}31104^{3}46656^{1}193312^{2} |

### 2.2 Linear Groups

In this section we provide words for the maximal subgroups of *L*_{2}(8), *L*_{2}(8) : 2, *L*_{2}(13), *L*_{2}(13) : 2, *L*_{2}(16), *L*_{2}(17), *L*_{2}(17) : 2, *L*_{2}(19), *L*_{2}(19) : 2, *L*_{2}(23), *L*_{2}(29), *L*_{2}(31), *L*_{3}(3), *L*_{3}(3) : 2, *L*_{3}(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 2^{3} : 7 of *L*_{2}(8), 9 : 6 and *L*_{2}(8) of the group *L*_{2}(8) : 2, *D*_{14} of the group *L*_{2}(13), *D*_{28} of the group *L*_{2}(13) : 2, *D*_{32} and *D*_{36} of the group *L*_{2}(17) : 2, *D*_{20} of the group *L*_{2}(19), *D*_{36} and 19 : 18 of the group *L*_{2}(19) : 2, *D*_{24} of the group *L*_{2}(23), *A*_{5} of the group *L*_{2}(29), *D*_{30}, *D*_{32} and *S*_{4} of the group *L*_{2}(31), 3^{2} : 2*S*_{4} of the group *L*_{3}(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 *L*_{2}(31) is computed by taking the normalizer of 31*AB i.e*., *N*(31*AB*) = 31 : 15. Similarly *N*(2*A*) = *D*_{32} and *N*(3*A*) = *D*_{30}. The normalizer here is computed by the methods given in [43] and the programmes given by simon [23] with a little change in them.

SubGroups | 1^{st}generator | 2^{nd}generator |
---|---|---|

D_{18} | b^{2}ab^{−1}(ab)^{2}(ab^{−1})^{2}aba | (ba)^{2}(b^{−1}a)^{2}(ba)^{2}b^{−1}a |

D_{14} | ab | b(abab^{−1})^{3}ab^{−1} |

2^{3} : 7 | a | b(abab^{−1})^{2}a |

Following Atlas, the subgroup *D*_{18} is the normalizer of 3*A*. *i.e*., *D*_{18} = *N*(3*A*). Similarly *D*_{14} = *N*(7*ABC*) and 2^{3} : 7 = *N*(2*A*^{3}).

The subgroups 9 : 6, 7 : 6 and *L*_{2}(8) were computed by random rearching, while 2^{3}.7 : 3 is computed by the information given in Atlas [34], *i.e*., 2^{3}.7 : 3 = *N*(2*A*^{3}).

Following Atlas, *D*_{14} = *N*(7*ABC*), *D*_{12} = *N*(2*A*), *A*_{4} = *N*(2*A*^{2}) and 13 : 6 = *N*(13*AB*).

SubGroups | 1^{st}generator | 2^{nd}generator |
---|---|---|

9 : 6 | a | b(abab^{−1})^{2}a |

7 : 6 | b | a^{(}(ab)^{3}bab^{2}(ab)^{2}b(ab)^{3}b(ab)^{5}) |

l_{2}(8) | a | ((ab)^{3}bab^{2})^{(}(ab)^{3}bab^{2}(ab)^{2}b(ab)^{3}b(ab)^{4}) |

2^{3}.7 : 3 | b^{2}ab^{−1}(ab)^{2}(ab^{−1})^{2}aba | (ba)^{2}(b^{−1}a)^{2}(ba)^{2}b^{−1}a |

SubGroups | 1^{st}generator | 2^{nd}generator |
---|---|---|

D_{14} | a | (ab)^{2}b |

D_{12} | (bab^{−1}a)^{2}(ba)^{2} | b(ab^{−1})^{2}a |

A_{12} | (bab^{−1}a)^{2}(ba)^{2} | (b^{−1}abab^{−1}a)^{2} |

13 : 6 | ab | b(ab^{−1})^{5}(ab)^{2}a |

SubGroups | 1^{st}generator | 2^{nd}generator |
---|---|---|

S_{4} | b | (ba)^{3}(b^{−1}a)^{2}b^{−1} |

D_{28} | a | ab^{−1}ab(ba)^{2} |

D_{24} | b^{−1}a(bab^{−1}ab)^{2}ab^{2}a | bab^{2}(ab^{−1})^{3}(ab)^{2}a |

L_{2}(13) | bab^{2}ab^{−1} | ba(bab)^{2}ab^{−1}(ab)^{2} |

The maximal subgroup *D*_{28} is computed by random rearching, while the remaining four were constructed by the information given in Atlas *D*_{28} = *C*(2*B*), *D*_{24} = *N*(3*A*) and *S*_{4} = *N*(2*A*^{2}).

SubGroups | 1^{st}generator | 2^{nd}generator |
---|---|---|

D_{30} | ab | ab^{−1}(ab)^{3}(ab^{−1})^{3}aba |

D_{34} | abab^{−1} | (bab^{−1}a)^{4}bab^{−1} |

A_{5} | b | (bab^{−1}a)^{4}bab^{−1} |

2^{4} : 15 | ab | (bab^{−1}a)^{4}bab^{−1} |

The maximal subgroup *A*_{5} is computed by random rearching, while the remaining maximal subgroups were constructed by the information given in Atlas *D*_{34} = *N*(17*A* − *H*), *D*_{30} = *N*(3*A*) and 2^{4} : 15 = *N*(2*A*^{4}).

SubGroups | 1^{st}generator | 2^{nd}generator |
---|---|---|

17 : 8 | ab | (ba)^{2}(b^{−1}a)^{2}(ba)^{3} |

D_{18} | (bab^{−1}a)^{2}bab^{−1} | (bab^{−1}a)^{2} |

D_{16} | (ba)^{3}(b^{−1}a)^{3}(ba)^{2} | ((b^{−1}a)^{2}ba)^{2}b^{−1}ab |

S_{4} | (abab^{−1}(ab)^{2})^{2}ab | abab^{−1}(ab^{−1}ab)^{2}abab^{−1}a |

The maximal subgroups were constructed by the information given in Atlas, *i.e*., 17 : 8 = *N*(17*AB*), *S*_{4} = *N*(2*A*^{2}), *D*_{18} = 3*A* and *D*_{16} = *N*(2*A*).

SubGroups | 1^{st}generator | 2^{nd}generator |
---|---|---|

D_{32} | a | ((ab)^{4}b)^{4} |

D_{36} | a | ((ba)^{4}b^{−1}a)^{2}b |

L_{2}(17) | b | (ababb)^{3} |

17 : 16 | (ab)^{6}ab^{−1}ab^{−1} | (b^{−1}a)^{5}(ba)^{3}b^{−1}ab^{−1} |

The maximal subgroups *D*_{32}, *L*_{2}17 and *D*_{36} were computed by random rearching, while the subgroup 17 : 16 is constructed by the information given in Atlas *i.e*., 17 : 16 = *N*(17*AB*).

SubGroups | 1^{st}generator | 2^{nd}generator |
---|---|---|

19 : 9 | ab | bab^{−1}(ab)^{2}(abab^{−1})^{2}(ab^{−1})^{2} |

D_{20} | a | b(ab^{−1})^{3}(ab)^{3}ab^{−1} |

D_{18} | abab^{−1} | (abab^{−1})^{4}a |

A_{5} | b | b(ab^{−1})^{3}(ab)^{3}ab^{−1} |

Following Atlas, 19 : 9 = *N*(19*AB*), *A*_{5} = *N*(2*A*, 3*A*, 5*AB*), *D*_{20} = *N*(2*A*) and *D*_{18} = *N*(3*A*).

SubGroups | 1^{st}generator | 2^{nd}generator |
---|---|---|

L_{2}(19) | b | (ab)^{2} |

D_{40} | ab | (ab)^{3}(ab(ab^{−1})^{2})^{2}(ab)^{2}ab^{−1}(ab)^{2}a |

D_{36} | a | (ba)^{7}(b^{−1}a)^{2}b(ab^{−1})^{3}aba |

S_{4} | b | b^{−1}ab^{−1}(ab^{−1a}b)^{2}(ab)^{2} |

19 : 18 | a | (ab^{−1}(ab)^{2})^{2}abab |

The maximal subgroups *L*_{2}(19), *D*_{36}, *S*_{4} and 19 : 18 were computed by random rearching, while the subgroup *D*_{40} is constructed by the information given in Atlas *i.e*., *D*_{40} = *N*(5*AB*).

SubGroups | 1^{st}generator | 2^{nd}generator |
---|---|---|

23 : 11 | ab | bab^{−1}(ab^{−1}ab)^{2}(ab^{−1})^{2}(ab)^{2}a |

D_{24} | a | (bab^{−1}a)^{4}bab^{−1} |

S_{4} | (ba)^{5}(b^{−1}a)^{2}b(abab^{−1})^{2} | (ba)^{5}(b^{−1}a)^{4} |

D_{22} | ((ab)^{2}ab^{−1})^{2} | (b^{−1}aba)^{2}b(ab^{−1})^{4}abab^{−1}a |

The maximal subgroups *D*_{22}), *D*_{24}, *S*_{4} and 23 : 11 was constructed by the information given in Atlas *i.e*., *D*_{22} = *C*(2*B*), *D*_{24} = *N*(2*A*), *S*_{4} = *N*(2*A*^{2}) and 23 : 11 = *N*(23*AB*).

SubGroups | 1^{st}generator | 2^{nd}generator |
---|---|---|

29 : 14 | ab | (ba)^{3}b(ab^{−1}abab^{−1})^{2}ab^{−1} |

D_{28} | (ba)^{2}(b^{−1}a)^{2}(ba)^{2}b^{−1}ab^{−1} | (ba)^{2}b^{−1}(ab)^{3}(ab^{−1})^{2}a |

D_{30} | ((ab)^{2}ab^{−1})^{2} | (ba)^{2}(b^{−1}a)^{2}(ba)^{2}b^{−1}ab^{−1} |

A_{5} | a | b^{−1}ab^{−1}(ab)^{7}ab^{−1}(ab)^{2}a |

The maximal subgroup *A*_{5} is computed by random rearching, while the subgroups *D*_{28}, *D*_{30} and 29 : 14 were constructed by the information given in Atlas *i.e*., *D*_{28} = *N*(2*A*), *D*_{30} = *N*(3*A*) and 29 : 14 = *N*(29*AB*).

SubGroups | 1^{st}generator | 2^{nd}generator |
---|---|---|

31 : 15 | ab | ab^{−1}ab(ab^{−1})^{4}ab(ab^{−1})^{2} |

D_{30} | a | bab^{−1} |

D_{32} | a | b^{−1}(ab^{−1}(ab)^{2})^{2}(ab)^{4}ab^{−1}aba |

A_{5} | b | b^{−1}(ab^{−1}(ab)^{2})^{2}(ab)^{4}ab^{−1}aba |

S_{4} | a | b^{2}(ab^{2})^{2}(ab)^{2} |

The maximal subgroups *D*_{30}, *D*_{32} and *S*_{4} were computed by random rearching, while the subgroup *A*_{5}, and 31 : 15 were constructed by the information given in Atlas *i.e*., *A*_{5} = *N*(2*A*, 3*A*, 5*AB*), and 31 : 15 = *N*(31*AB*).

SubGroups | 1^{st}generator | 2^{nd}generator |
---|---|---|

3^{2} : 2S_{4} | a | ab^{2}(ab)^{2} |

S_{4} | abab^{−1}((ab^{−1})^{2}(ab)^{2})^{2}abab^{−1} | ((ab^{−1})^{2}(ab)^{2})^{2}(ab^{−1})^{3} |

13 : 3 | ab | (ba)^{6}b^{−1}(ab)^{2}ab^{−1}aba |

The maximal subgroups 3^{2} : 2*S*_{4}, *S*_{4} and 13 : 3 were constructed by the information given in Atlas *i.e*., 3^{2} : 2*S*_{4} = *N*(3*A*^{2}), *S*_{4} = *N*(2*A*^{2}) and 13 : 3 = *N*(13*ABCD*).

SubGroups | 1^{st}generator | 2^{nd}generator |
---|---|---|

L_{3}(3) | (b^{−1}aba)^{2} | ((ab^{−1})^{2}(ab)^{2})^{2}(ab^{−1})^{3} |

S_{4} : 2 | (b^{2}a)^{4}(b^{−1}a)^{2}b^{−1}(ab^{2})^{2}(ab)^{3} | b^{2}(ab^{−1})^{5}abab^{−1}(ab)^{2} |

2.S_{4}.2 | ((ab)^{2}a)^{2}b | (ab)^{3}(ba)^{2}b((ba)^{2}b^{2})^{6}((ba)^{2}b^{2}a)^{9} |

3^{1+2}.D_{8} | a(bab)^{2}ab | (ba)^{3}(b^{−1}a)^{2}b^{−1}(ab^{2})^{2}a |

13 : 6 | (ab)^{2}a(ab)^{4} | (ab)^{3}(ba)^{2}b((ba)^{2}b^{2})^{2}a |

The maximal subgroups *L*_{3}(3), 2.*S*_{4}.2, 3^{1+2}.*D*_{8} and 13 : 6 were constructed by random searching, while the subgroup *S*_{4} : 2 is constructed by the information given in Atlas *i.e*., *S*_{4} : 2 = *C*(2*B*).

SubGroups | 1^{st}generator | 2^{nd}generator |
---|---|---|

5^{2} : GL_{2}(5) | bab(ba)^{3}b^{−2}a | b^{−1}aba^{−1}b^{−1}a^{−1}(bab)^{2}(ab^{−1})^{2}b^{−1}a |

S_{5} | (ab)^{2}a(ab)^{5} | (ab)^{3}(ba)^{2}b(ba)^{45}((ba)^{3}b)^{3} |

4^{2} : S_{3} | (ab)^{2}a(ab)^{5} | (ab)^{3}((ba)^{2}b)^{3}(ba)^{21}((ba)^{3}b)^{7} |

S_{5} | (ab)^{2}a(ab)^{5} | (ab)^{3}((ba)^{2}b)^{5}(ba)^{48}((ba)^{3}b)^{8} |

The maximal subgroups 5^{2} : *GL*_{2}(5), *S*_{5}, and 4^{2} : *S*_{3} were constructed by the information given in Atlas *i.e*., 5^{2} : *GL*_{2}(5) = *N*(5*A*^{2}), *S*_{5} = *N*(2*A*, 3*A*, 5*AB*), and 4^{2} : *S*_{3} = *N*(2*A*^{2}).

## 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 *S*_{5} in the list of maximal subgroups of *L*_{3}(5), but here we provide generators for two non-conjugate copies of *S*_{5}.

## 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.

**Data Availability Statement**: All data required for this research is available in this paper.**Ethical approval**: The conducted research is not related to either human or animal use.**Author Contribution**: All authors contribute equally in this paper.**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).**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.