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

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.


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 HSand 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  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.

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

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 ∼ = 3×10 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, )︂ 6 = 1 q − 6 + 134q + 760q 2 + 3345q 3 + 12256q 4 + 39350q 5 + .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]  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(2B 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.

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 3A and 11A classes of A 12 fuse to 3A and 11A 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 (2 2 ) 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 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 15 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 .

Generators of 2 3+2+6 .(3 × L 3 (2))
Here our required subgroup is the N HN (2B 3 ) [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 2 1+8 inside the centralizer of 2B-involution [42].
The group 2 1+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 2 1+8 . Now searching inside 2 1+8 , one can easily find a 2B 3 . The details of computing the 2B 3 are given below: The element of 2B 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 HNare given by: Thus generators of 2B 3 are b 7 = b 10 6 , b 8 = (b 1 b 2 b 4 ) 6 and b 9 = (b 1 b 3 b 5 ) 6 . The generators for normalizer of 2B 3 inside HNare given below.

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.
Now we give some random elements of 3 1+4 : 4.A 5 .  The generators of the normalizer of 3 4 inside HN are given below.
The orbit shapes and order of the above subgroups of HN are given below.

Linear Groups
In this section we provide words for the maximal subgroups of L 2 (8), L 2 (8)  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 The normalizer here is computed by the methods given in [43] and the programmes given by simon [23] with a little change in them.  (8).
Following Atlas, the subgroup D 18 is the normalizer of 3A. i.e., D 18  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., The maximal subgroup D 28 is computed by random rearching, while the remaining four were constructed by the information given in Atlas D 28   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

SubGroups
1 st generator 2 nd generator D 32 a 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(17AB).