Words for maximal Subgroups of Fi24‘

Abstract Group Theory is the mathematical application of symmetry to an object to obtain knowledge of its physical properties. The symmetry of a molecule provides us with the various information, such as - orbitals energy levels, orbitals symmetries, type of transitions than can occur between energy levels, even bond order, all that without rigorous calculations. The fact that so many important physical aspects can be derived from symmetry is a very profound statement and this is what makes group theory so powerful. In group theory, a finite group is a mathematical group with a finite number of elements. A group is a set of elements together with an operation which associates, to each ordered pair of elements, an element of the set. In the case of a finite group, the set is finite. The Fischer groups Fi22, Fi23 and Fi24‘ are introduced by Bernd Fischer and there are 25 maximal subgroups of Fi24‘. It is an open problem to find the generators of maximal subgroups of Fi24‘. In this paper we provide the generators of 10 maximal subgroups of Fi24‘.


Introduction
Group Theory is the mathematical application of symmetry to an object to obtain knowledge of its physical properties, providing a quick and simple method to determine the relevant physical information about the molecule [1]. The symmetry of a molecule provides information on what the energy levels of the orbitals will be, what the orbitals symmetries are, what transitions can occur between energy levels, and even bond order to name a few can be found, all without rigorous calculations [2]. The fact that so many important physical aspects can be derived from symmetry is a very profound statement and this is what makes group theory so powerful [3,4].
The allocated point groups would then be able to be utilized to decide physical properties, (for example, concoction extremity and chirality), spectroscopic properties (especially valuable for Raman spectroscopy, infrared spectroscopy, round dichroism spectroscopy, mangnatic dichroism spectroscopy, UV/Vis spectroscopy, and fluorescence spectroscopy), and to build sub-atomic orbitals. Sub-atomic symmetry is in charge of numerous physical and spectroscopic properties of compounds and gives important data about how chemical reaction happen. So as to appoint a point group for some random atom, it is important to locate the arrangement of symmetry activities present on it [5,6]. The symmetry activity is an activity, for example, a rotation about an axis or a reflection through a mirror plane. It is a task that moves the particle with the end goal that it is undefined from the first setup. In group theory, the rotation axis and mirror planes are classified "symmetry components". These components can be a point, line or plane concerning which the symmetry task is done. The symmetry activities of a particle decide the particular point aggregate for this atom. Water atom with symmetry axis in chemistry, there are five critical symmetry tasks. The identity operation (E) comprises of leaving the atom as it is. This is equivalent to any number of full turns around any axis. This is a symmetry all things considered, though the symmetry group of a chiral atom comprises of just the indentity operation. Rotation around an axis (Cn) comprises of pivoting the atoms around a particular axis by a particular point. For instance, if a water particle turns 180° around the axis that goes through the oxygen molecule and between the hydrogen atoms, it is in same configration from it began. For this situation, n = 2, since applying it twice creates the identity operation. Other symmetry tasks are: reflection, reversal and improper revolution (rotation followed by reflection) [7,8,9].
In abstract algebra and mathematics group theory studies the algebraic structures known as groups [10,11,12,13]. The idea of a group is central to abstract algebra: other well-known algebraic structures, for example rings, fields, and vector spaces, are all able to be viewed as groups supplied with extra axioms and operations. Groups repeat all mathematics, and the techniques for group theory have affected numerous parts of algebra. Lie groups and linear abstract groups are two branches of group theory that have encountered progresses and have turned out to be branches of knowledge in their own right. Different physical frameworks, for example hydrogen atoms and the crystals, might be displayed by symmetry groups. In this manner group theory and the firmly related representation theory have numerous vital applications in physical science, general science and material sciences. Group theory is additionally key to open key cryptography [14,15,16,17,18,19,20]. The general numerical meaning of a group can be connected to phenomena occurring in a wide range of disciplines. A sporadic group is one of the 26 remarkable groups found in the characterization of finite simple groups. A simple group is a group, G that does not have any normal subgroups aside from the trivial group and G itself. The classification theorem says that the list of finite simple groups comprises of 18 countably infinite families, in addition to 26 exemptions that donot take after such a systematic pattern. These are the sporadic groups. They are otherwise called the sporadic basic groups, or the sporadic finite groups. Since it isnot entirely a group of Lie type, the Tits group is sometimes viewed as a sporadic group, in which case the sporadic group number 27.
The three sporadic simple groups are the Fischer groups introduced by Bernd Fischer in [21], and denoted as Fi 22 , Fi 23 and Fi 24 . Bernd Fischer discovered Fischer groups when he was investigating three-transposition groups and these groups have the following special properties: -Fischer groups are generated by conjugacy class of elements having order 2, which is called 3-transpositions or Fischer transpositions. -For any two distinct transpositions, the product has order 2 or 3.
Due to these interesting properties these groups have been studied extensively and many papers have been written on them. A typical example of a 3-transposition group is a symmetric group [22], where the Fischer transpositions are truly transpositions. The symmetric group S n can be generated by n-1 transpositions: (12), (23),..., (n-1,n). Fischer could characterize 3-transposition groups that fulfill certain additional specialized conditions. The groups he discovered fell generally into a few infinite classes (other than symmetric groups: symmetrical groups, unitary, and certain classes of symplectic groups), however he also discovered 3 large groups. These groups are generally alluded to as Fi 22  A beginning stage for the Fischer groups is the unitary gathering PSU 6 (2), which can be thought of as group among 3 Fischer groups, having order 9,196,830,720=2 15 .3 6 .5.7.11. In fact it is the double cover 2.PSU6(2) that turns into a subgroup of a new group. This is stabilizer of one vertex in a graph of 3510(=2. Chang Choi [23] found all the maximal subgroups of M 24 . The maximal subgroups of HS and McL groups were discovered by Spyros Magliveras [24] and Lerry Finkelstein [25]. Finkelstein then worked with Arunas Rudvalis to target the maximal subgroups of J 2 [26] and J 3 [27]. Gerard Enright completed his thesis in 1977 under the supervision of Conway, in which he discussed the subgroups of the Fischer groups Fi 22 and Fi 23 generated by transpositions [28]. The local and non-local subgroups of Fi 22 , Fi 23 and Fi 24 are given in [29]. In 1990, Steve Linton determined the maximal subgroups of Th, Fi 24 and its automorphism groups. He completely discussed the maximal subgroups in [30,31]. In 1999, Wilson constructed the maximal subgroups of B [32]. 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 and its standard generators are given in [33]. The concept of standard generators for sporadic simple groups was introduced by R. A. Wilson [34]. He started a project known as 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 22 . Words for maximal subgroups of these groups are given on world-wide-web [37].
The Fischer group Fi 24 , is one of the 26 sporadic simple groups which occur in the classification of finite simple groups. There are 25 maximal subgroups of the group Fi 24 ' [33,37]. The maximal subgroups of Fi 24 are given below.
Here we provide words for maximal subgroups which are marked by the * above. In some cases we have been able to reduce the length of words involved. The normalizers of certain subgroups which are the crux of the matter here were computed by the methods given in [35,38].We have used GAP [39] and MAGMA [40] for computations and our notation follows [37].

Methods to find words for the Maximal subgroups for Fischer group 2.1 Trawling
This method can be used to find the words for the maximal subgroups of sporadic simple group. This method works only for easy cases or large order subgroups where it is not possible to generate all the desired maximal subgroups. The generators found with this method are not standard generators and involves the following steps. 1) First we fix any element of G, say a∈G, we call this fixed element as the first generator of the desired maximal subgroup.
2) Now for each different element of G, say μ∈G if <a, μ>, is the required proper subgroup then stop otherwise go to the step 1.
The words usually found with this technique are short words. Using this technique we found words for Fi 23 (maximal subgroup of Fi 24 '). Words for Fi 23 are given in section 3.

Maximal subgroups by Conjugacy Classes
This method is often used when we are searching for the standard generators of the maximal subgroup. This method involves the following steps. 1. First we have to find out the right conjugacy classes in which our required maximal subgroup lies. 2. Next we find the corresponding conjugacy classes of the parent group by using class fusion from maximal subgroup to the parent group. The probability of finding the standard generators depends on the size of the conjugacy classes involved. Let G 1 and G 2 be the conjugacy classes of G with a,a 1 ∈G and b,b 1 ∈G the probability that (a, b, ab) is conjugate to (a 1 , b 1 , a 1 b 1 ) is . This method is more efficient when we are working with small conjugacy classes.Using this technique we found words for O 10 (2) (maximal subgroup of Fi 24 '). The computational detail is given in section 3.

Maximal subgroups by construction
This method is laboriousand involves different stages during construction while we are searching for the desired maximal subgroup. We have used this method only when the subgroup cannot be easily generated by random searching. Here we mention some of the cases which can be generated by construction.
The maximal subgroups which we are looking for usually occur as a normalizer or centralizer of some elementary abelian groups. The centralizer and normalizers are computed by the methods given in [18,21]. Here we mention those cases in which we need to calculate the normalizer or centralizer.

Main Results
In this section we provide words for 3 1+10

Construction of Fi 23 inside Fi 24 '
The words for Fischer group (Fi 23 ) are found by trawling. Before computing the words for Fi 23 we give some random elements.
The words for Fi 23 are given by a 2 and .

Construction of 3 1+10 :U 5 (2):2 inside Fi 24 '
From Atlas [37], the maximal subgroup under consideration is the normalizer of an element of conjugacy class 3A. This group is constructed in three steps summerized underneath.

Step 1
Firstly, we compute an element of class 3A. For that we give some words of Fi 24 ' given below.
At that point utilizing the power maps to discover a elements of order 3 and afterward check its centralizer order which affirms that the element belongs to class 3A i.e., .

Step 2
In this progression we will compute the normalizer of a 2 inside Fi 24 . The normalizer can be found by utilizing the method given in [18] for example we build the partial normalizer of a 2 inside various subgroups of Fi 24 . At that point consolidating these incomplete normalizers to get the necessary normalizer. The calculations of these normalizers are given beneath. Consider the group , then compute the normalizer of a 2 inside H 1 . Before computations, we will provide some words of H 1 , which will facilitate the computations. These words are given by Next we use the "TKnormalizertest" given by Simon [15] to compute the words for the partial normalizer of a 2 inside H 1 . These elements of the normalizer of a 2 inside H 1 is given by Find an involution inside the above calculated partial normalizer. This involution is given by . Now consider and in a similar way, we compute normalizer of a 2 inside H 2 .
The words for the partial normalizer are given by.
Find the centralizer of g 6 . The words for the Centralizer of g 6 are given by.
Next we find the partial normalizer of a 2 inside the above calculated entralizer. The words for the partial normalizer are given below. Now consolidating the above normalizers will gives us the words for 3 1+10 :U 5 (2):2 given by g 1 g 2 and g 7 .

Construction of O 10 (2) inside Fi 24 '
Following Atlas we found this group inside the conjugacy classes of 2A, 7A and 11A. Before computations we give some random elements.
Then using power maps the elements of class 7A and 11A are given by x 8 and x 9 respectively.
The words for O 10 (2) are given by a and .

.O 7 (3)=N(3 7 ).
Here first we find an element of order 3. This element is given by a 1 =((ab) 4 b(ab) 6 b) 4 . Then we find the centralizer of a 1 inside Fi 24 '. This can be constructed by the technique given in [18].
The generators for the elementary abelian group of order 2187 are given by Next we just compute the normalizer of which is the required maximal subgroup 3 7 .O 7 (3). The normalizer can be computed by the method given in [37] i.e., first we find a single involution and then searching inside this involution centralizer the partial normalizer of H The involution is given by . The generators for the Centralizer of l 2 are given below.
It just remains to calculate the normalizer of a 1 . This can be computed by using the method given in [31]. The generators for the centralizer of a 1 are given below.

Construction of 2 11 .M 24 inside Fi 24 '
Following Atlas required group (2 11 .M 24 ) is the normalizer of 2 11 .First we find an involution then searching inside its centralizer will gives us 2 11 .This element is given by a 1 =((ab) 4 b) 18 . The generators for the centralizer of a 1 are given below. Then find an involution (d 1 ) inside and then compute the centralizer of d 1 inside H 3 . The generators for the centralizer of d 1 are given below. Now the generators for 2 11 can easily be found from the above calculated centralizer given below.
We just compute the normalizer of 2 11 which turns out 2 11 . M 24 . The words for 2 11 .M 24 are given by a and ((ab) 3

Construction of (3×O 8 (3):3):2 inside Fi 24 '
Following Atlas [37], required maximal subgroup (3×O 8 (3):3):2=N(3A). The element of class 3A is given by a 3 =(ababababababb) 20 . It just remains to calculate the normalizer which can be computed by the method given in [38] i.e., we compute the partial normalizers and then combining these partial normalizers to get the required one. Before computations we will give some random words given by.
Here a 5 is an involution we will found the centralizer of that involution. The generators for the centralizer of a 5 are given below. Now we wil find the partial normalizer of a 3 inside using the programes given in [24]. The words for the partial normalizer are given below.
Next we found an involution inside the above calculated partial normalizer. This involution is given by , then searching the partial normalizer of a 3 inside the centralizer of a 6 . the generators for the centralizer of a 6 are given below.
The words for the partial normalizer of a 3 inside are given by.
Combining the above calculated partial normalizers we get the required normalizer of ((

Appliactions in Chemistry
Symmetric groups fund many applications in chemistry, material sciences and physics. Many books and research papers have been written on appllications of goup thoery in chemistry because a unit molecule is a fundamental unit from which pure substance is constructed, and can be assigned a symmetric group, as most of the substances are symmetric [42].For example, the density matrix renormalization group is a method that is useful for describing molecules that have strongly correlated electrons. In [43], authors provided a pedagogical overview of the basic challenges of strong correlation, how the density matrix renormalization group works. According to [44], all structural formulas of covalently bonded compounds are graphs: they are therefore called molecular graphs or, better constitutional graphs and every graph can be studied with the help of associated group. The group theoretical studies of molecular structure has strong relationship with quantum mechanics [45]. The general quantum-mechanical perturbation theory on the nonlinear optical effect in crystals and gives a systematic presentation of the basic concepts and calculation methods of the 'anionic group theory for the nonlinear optical effect of crystals [46]. Group theory is used for calculation of equilibrium geometries, normal mode vibrational frequencies, reaction energies, electric dipole moments [47] and electron transfer reactions [48]. The properties of inorganic and organometallic compounds [49] and vibrational spectroscopy of carboranes and parent boranes [50] can be studied with the help of group theory.

Conclusions and future work
Molecular symmetry in chemistry portrays the symmetry present in molecular and the order of particles as indicated by their symmetry. Atomic symmetry is an essential idea in chemistry, as it tends to be utilized to anticipate or clarify a significant number of a particle's compound properties, for example, its dipole minute and its permitted spectroscopic advances. Numerical instruments are utilized to portray the symmetry of molecles [51][52][53][54][55][56][57][58][59].
Group Theory is the numerical use of symmetry to an object to get information of its physical properties. What group hypothesis conveys to the table, is the manner by which the symmetry of a molecule is identified with its physical properties and gives a brisk basic technique to decide the significant physical data of the molecule. The symmetry of a molecule gives us its properties.. The way that such a large number of vital physical perspectives can be gotten from symmetry is an extremely significant articulation and this is the thing that makes group theory so incredible. In this paper we provide words for the 10 maximal subgroups of the Fischer group Fi 24 ', however there are still some cases of Fi 24 ', Baby Monster Group (B) and Monster Group (M).