Open Chem., 2019; 17: 1491–1500
Research Article
Faisal Yasin, Adeel Farooq, Chahn Yong Jung*
Words for maximal Subgroups of Fi24‘ https://doi.org/10.1515/chem-2019-0156 received October 16, 2018; accepted December 1, 2019. energy levels, and even bond order to name a few can be found, all without rigorous calculations [2]. The fact that Abstract: Group Theory is the mathematical application so many important physical aspects can be derived from of symmetry to an object to obtain knowledge of its symmetry is a very profound statement and this is what physical properties. The symmetry of a molecule provides makes group theory so powerful [3,4]. us with the various information, such as - orbitals energy The allocated point groups would then be able to levels, orbitals symmetries, type of transitions than can be utilized to decide physical properties, (for example, occur between energy levels, even bond order, all that concoction extremity and chirality), spectroscopic without rigorous calculations. The fact that so many properties (especially valuable for Raman spectroscopy, important physical aspects can be derived from symmetry infrared spectroscopy, round dichroism spectroscopy, is a very profound statement and this is what makes mangnatic dichroism spectroscopy, UV/Vis spectroscopy, group theory so powerful. In group theory, a finite group and fluorescence spectroscopy), and to build sub-atomic is a mathematical group with a finite number of elements. orbitals. Sub-atomic symmetry is in charge of numerous A group is a set of elements together with an operation physical and spectroscopic properties of compounds and which associates, to each ordered pair of elements, an gives important data about how chemical reaction happen. element of the set. In the case of a finite group, the set is So as to appoint a point group for some random atom, it is finite. The Fischer groups Fi22, Fi23 and Fi24‘ are introduced important to locate the arrangement of symmetry activities by Bernd Fischer and there are 25 maximal subgroups present on it [5,6]. The symmetry activity is an activity, for of Fi24‘. It is an open problem to find the generators of example, a rotation about an axis or a reflection through maximal subgroups of Fi24‘. In this paper we provide the a mirror plane. It is a task that moves the particle with the generators of 10 maximal subgroups of Fi24‘. end goal that it is undefined from the first setup. In group theory, the rotation axis and mirror planes are classified Keywords: molecular symmetry; point group; Maximal “symmetry components”. These components can be a subgroup; Fischer group; generators. 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 1 Introduction symmetry axis in chemistry, there are five critical symmetry tasks. The identity operation (E) comprises of leaving the Group Theory is the mathematical application of symmetry atom as it is. This is equivalent to any number of full turns to an object to obtain knowledge of its physical properties, around any axis. This is a symmetry all things considered, providing a quick and simple method to determine the though the symmetry group of a chiral atom comprises relevant physical information about the molecule [1]. The of just the indentity operation. Rotation around an axis symmetry of a molecule provides information on what (Cn) comprises of pivoting the atoms around a particular the energy levels of the orbitals will be, what the orbitals axis by a particular point. For instance, if a water particle symmetries are, what transitions can occur between 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 *Corresponding author: Chahn Yong Jung, Department of Business Administration, Gyeongsang National University, Jinju 52828, South symmetry tasks are: reflection, reversal and improper Korea, E-mail: [email protected] revolution (rotation followed by reflection) [7,8,9]. Faisal Yasin, Adeel Farooq, Department of Mathematics, COMSATS In abstract algebra and mathematics group theory University Islamabad, Lahore Campus, Pakistan studies the algebraic structures known as groups Department of Mathematics and Statistics, The University Faisal Yasin, [10,11,12,13]. The idea of a group is central to abstract algebra: of Lahore, Lahore 54000, Pakistan
Open Access. © 2019 Faisal Yasin et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. 1492 Faisal Yasin et al. other well-known algebraic structures, for example however he also discovered 3 large groups. These groups rings, fields, and vector spaces, are all able to be viewed are generally alluded to as Fi22, Fi23 and Fi24. The Fi22, Fi23 as groups supplied with extra axioms and operations. are simple groups while Fi24 contains the simple group
Groups repeat all mathematics, and the techniques for Fi‘24 of index 2. group theory have affected numerous parts of algebra. A beginning stage for the Fischer groups is the unitary
Lie groups and linear abstract groups are two branches gathering PSU6 (2), which can be thought of as group among of group theory that have encountered progresses and 3 Fischer groups, having order 9,196,830,720=215.36.5.7.11. have turned out to be branches of knowledge in their In fact it is the double cover 2.PSU6(2) that turns into a own right. Different physical frameworks, for example subgroup of a new group. This is stabilizer of one vertex in hydrogen atoms and the crystals, might be displayed a graph of 3510(=2.33.5.13). These vertices are recognized by symmetry groups. In this manner group theory and as conjugate 3-transpositions in Fi22 of the graph. the firmly related representation theory have numerous The Fischer groups are named by similarity with vital applications in physical science, general science the substantial Mathieu groups. A maximal set, in Fi22 of and material sciences. Group theory is additionally 3-transpositions all computing with each other has size 22 key to open key cryptography [14,15,16,17,18,19,20]. The and is known as a basic set. There are 1024 3-transpositions, general numerical meaning of a group can be connected that donot compute with any in the specific basic set called to phenomena occurring in a wide range of disciplines. anabasic. Any of other 2364, computes with 6 basic sets is A sporadic group is one of the 26 remarkable groups called hexadic. The arrangements of 6 produce a S(3,6,22) found in the characterization of finite simple groups. A called a Steiner system, which contains the symmetric simple group is a group, G that does not have any normal group M22. An abelian group of order 210 is generated by subgroups aside from the trivial group and G itself. The a basic set which stretches out in to a subgroup 210:M22. classification theorem says that the list of finite simple Chang Choi [23] found all the maximal subgroups of groups comprises of 18 countably infinite families, in M24. The maximal subgroups of HS and McL groups were addition to 26 exemptions that donot take after such a discovered by Spyros Magliveras [24] and Lerry Finkelstein systematic pattern. These are the sporadic groups. They [25]. Finkelstein then worked with Arunas Rudvalis to are otherwise called the sporadic basic groups, or the target the maximal subgroups of J2 [26] and J3 [27]. Gerard sporadic finite groups. Since it isnot entirely a group of Enright completed his thesis in 1977 under the supervision Lie type, the Tits group is sometimes viewed as a sporadic of Conway, in which he discussed the subgroups of the group, in which case the sporadic group number 27. Fischer groups Fi22 and Fi23 generated by transpositions
The three sporadic simple groups are the Fischer [28]. The local and non-local subgroups of Fi22, Fi23 and groups introduced by Bernd Fischer in [21], and denoted as Fi24 are given in [29]. In 1990, Steve Linton determined
Fi22, Fi23 and Fi24. Bernd Fischer discovered Fischer groups the maximal subgroups of Th, Fi24 and its automorphism when he was investigating three-transposition groups and groups. He completely discussed the maximal subgroups these groups have the following special properties: in [30,31]. In 1999, Wilson constructed the maximal –– Fischer groups are generated by conjugacy subgroups of B [32]. Recently Wilson has updated the list class of elements having order 2, which is called of maximal subgroups of the Monster Group. There are still 3-transpositions or Fischer transpositions. some undetermined cases. To date there are 44 maximal –– For any two distinct transpositions, the product has subgroups of Monster and its standard generators are order 2 or 3. given in [33]. The concept of standard generators for sporadic simple groups was introduced by R. A. Wilson Due to these interesting properties these groups have been [34]. He started a project known as online version of Atlas, studied extensively and many papers have been written which would provide not only representations (matrix and on them. A typical example of a 3-transposition group is permutation) but also words for the maximal subgroups a symmetric group [22], where the Fischer transpositions of simple and almost simple groups. The words for the are truly transpositions. The symmetric group Sn can be maximal subgroups of M12.2, M22.2, HS.2, McL.2, J2.2, Suz.2, generated by n-1 transpositions: (12),(23),..., (n-1,n). He.2, Fi22.2, HN.2 and Fi24 are dicussed in [35]. In 2001, John Fischer could characterize 3-transposition groups N. Bray worked on the maximal subgroups of sporadic that fulfill certain additional specialized conditions. The simple groups of order less than 1014. In [36] he presents groups he discovered fell generally into a few infinite a complete list by providing words for the maximal classes (other than symmetric groups: symmetrical subgroups of 17 sporadic simple groups which includes groups, unitary, and certain classes of symplectic groups), M11, M12, J1, M22, J2, J3, Ru, O‘N, Co3, HS, McL, Suz, He, Fi24, Words for maximal Subgroups of Fi24‘ 1493
∈ Co2, M24, M23 and Fi22. Words for maximal subgroups of 1) First we fix any element of G, say a G, we call this these groups are given on world-wide-web [37]. fixed element as the first generator of the desired maximal
The Fischer group Fi24, is one of the 26 sporadic simple subgroup. groups which occur in the classification of finite simple 2) Now for each different element of G, say μ∈G if
[33,37]. The maximal subgroups of Fi24 are given below. go to the step 1. The words usually found with this technique are
short words. Using this technique we found words for Fi23
(maximal subgroup of Fi24‘). Words for Fi23 are given in section 3.
2.2 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 G1 and G2 be
the conjugacy classes of G with a,a1∈G and b,b1∈G the
probability that (a, b, ab) is conjugate to (a1, b1, a1 b1) Here we provide words for maximal subgroups which are is . This method is more efficient when marked by the * above. In some cases we have been able we are working with small conjugacy classes.Using to reduce the length of words involved. The normalizers of this technique we found words for O10 (2) (maximal certain subgroups which are the crux of the matter here subgroup of Fi24‘). The computational detail is given were computed by the methods given in [35,38].We have in section 3. used GAP[39] and MAGMA [40] for computations and our notation follows [37]. 2.3 Maximal subgroups by construction
This method is laboriousand involves different stages 2 Methods to find words for the during construction while we are searching for the desired Maximal subgroups for Fischer maximal subgroup. We have used this method only when the subgroup cannot be easily generated by random group searching. Here we mention some of the cases which can be generated by construction. 2.1 Trawling The maximal subgroups which we are looking for usually occur as a normalizer or centralizer of This method can be used to find the words for the maximal some elementary abelian groups. The centralizer and subgroups of sporadic simple group. This method works normalizers are computed by the methods given in only for easy cases or large order subgroups where it is not [18,21]. Here we mention those cases in which we need possible to generate all the desired maximal subgroups. to calculate the normalizer or centralizer. The maximal 1+10 The generators found with this method are not standard subgroup 3 :U5 (2):2 is the normalizer of an element of 1+10 7 7 generators and involves the following steps. class 3A i.e., 3 :U5 (2):2=N(3A). Similarly 3 .O7 (3)=N(3 ), 1494 Faisal Yasin et al.
1+12 11 11 2 :3.U4 (3).2=N(2B), 2 .M24=N(2 ), 2.Fi2.2=N(2A), (3×O8 At that point utilizing the power maps to discover a 2 2 (3):3):2=N(3A ) and (A4×O8 (2):3):2=N(2A ). elements of order 3 and afterward check its centralizer order which affirms that the element belongs to class 3A i.e., . 3 Main Results
1+10 1+10 3.2.2 Step 2 In this section we provide words for 3 :U5 (2):2, 3 :U5 (2):2, 37.O (3), 21+12:3.U (3).2, 211.M , 2.Fi .2, (3×O (3):3):2, 7 4 24 22 8 In this progression we will compute the normalizer of a Fi , O (2) and (A ×O (2):3):2. In some cases we reduce the 2 23 10 4 8 inside Fi . The normalizer can be found by utilizing the length of the words where possible. 24 method given in [18] for example we build the partial
normalizer of a2 inside various subgroups of Fi24. At that point consolidating these incomplete normalizers to 3.1 Construction of Fi inside Fi ‘ 23 24 get the necessary normalizer. The calculations of these normalizers are given beneath. The words for Fischer group (Fi ) are found by trawling. 23 Consider the group , then compute Before computing the words for Fi we give some random 23 the normalizer of a inside H . Before computations, we elements. 2 1 will provide some words of H1, 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 a2 inside The words for Fi23 are given by a2 and . H1. These elements of the normalizer of a2 inside H1 is given by
1+10 3.2 Construction of 3 :U5 (2):2 inside Fi24‘
From Atlas [37], the maximal subgroup under consideration Find an involution inside the above calculated partial is the normalizer of an element of conjugacy class 3A. normalizer. This involution is given by . Now This group is constructed in three steps summerized consider and in a similar way, we compute underneath. normalizer of a2 inside H2.
3.2.1 Step 1
Firstly, we compute an element of class 3A. For that we give some words of Fi24‘ given below.
The words for the partial normalizer are given by. Words for maximal Subgroups of Fi24‘ 1495
2+4+8 3.4 Construction of 3 .(A5×2A4).2 and 7 3 .O7 (3) inside Fi24‘
7 7 Following Atlas 3 .O7 (3)=N(3 ). Here first we find an 4 Find the centralizer of g6. The words for the Centralizer of element of order 3. This element is given by a1=((ab) 6 4 g6 are given by. b(ab) b) .
Then we find the centralizer of a1 inside Fi24‘. This can be constructed by the technique given in [18].
Next we find the partial normalizer of a2 inside the above calculated entralizer. The words for the partial normalizer are given below. The generators for the elementary abelian group of order 2187 are given by
Now consolidating the above normalizers will gives us the 1+10 words for 3 :U5 (2):2 given by g1 g2 and g7.
3.3 Construction of O10(2) inside Fi24‘ Next we just compute the normalizer of Following Atlas we found this group inside the conjugacy which is the required 7 classes of 2A, 7A and 11A. Before computations we give maximal subgroup 3 .O7 (3). The normalizer can be some random elements. 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 l2 are given below. Then using power maps the elements of class 7A and 11A are given by x8 and x9 respectively.
The words for O10(2) are given by a and . 1496 Faisal Yasin et al.
Then find an involution (a6) inside and then compute the
centralizer of a6 inside H. The generators for the centralizer
of a6 are given below.
2+4+8 The words for the maximal subgroup of 3 .(A5×2A4).2 7 are l1 l6 l8 and l3 l7 and the words for 3 .O7 (3) are l1 l6 l8 and l3 l7 l11. Then find an involution (b3) inside and then compute the
centralizer of b3 inside H1. The generators for the centralizer 1+12 3.5 Construction of 2 :3.H4(3).2 inside Fi24‘ of b3 are given below.
1+12 Following Atlas required group 2 :3.H4(3).2 is the normalizer of an element of class 2B which can easily be found by using the power map. This element is given by 4 18 a1=((ab) b) .
It just remains to calculate the normalizer of a1. This can be computed by using the method given in [31]. The generators for the centralizer of a1 are given below.
Then find an involution (c3) inside and then compute the
centralizer of c3 inside H2. The generators for the centralizer
of c3 are given below.
1+12 The generators for 2 :3.U4(3).2 are a3 and a4.
11 3.6 Construction of 2 .M24 inside Fi24‘
11 Following Atlas required group (2 .M24) is the normalizer of 211.First we find an involution then searching inside its centralizer will gives us 211.This element is given by 4 18 a1=((ab) b) . The generators for the centralizer of a1 are given below. Then find an involution (d1) inside and
then compute the centralizer of d1 inside H3. The generators
for the centralizer of d1 are given below. Words for maximal Subgroups of Fi24‘ 1497
one. Before computations we will give some random words given by.
Now the generators for 211 can easily be found from the above calculated centralizer given below. Here a5 is an involution we will found the centralizer of
that involution. The generators for the centralizer of a5 are given below.
We just compute the normalizer of 211 which turns out 211. 11 3 4 M24. The words for 2 .M24 are given by a and ((ab) b(ab) Now we wil find the partial normalizer of a3 inside b(ab)3 b(ab)5 b)7. using the programes given in [24]. The words for the partial normalizer are given below.
3.7 Construction of 2.Fi22.2 inside Fi24‘
Following Atlas 2.Fi22.2=N(2A). The element of class 2A is given by a. The centralizer of a can be computed by the method given in [31]. The generators for the centralizer of a 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 a3 inside the
centralizer of a6. the generators for the centralizer of a6 are given below.
The generators of the required group 2.Fi22.2 are a1 and a4.
3.8 Construction of (3×O8 (3):3):2 inside Fi24‘
The words for the partial normalizer of a3 inside
Following Atlas [37], required maximal subgroup (3×O8 are given by. (3):3):2=N(3A). The element of class 3A is given by 20 a3=(ababababababb) . 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 1498 Faisal Yasin et al.
Combining above two partial normalizers. The generators for (3×O8 (3):3):2 are k4, k5 and k6.
3.9 Construction of ((A4×O8 (2):3):2) inside
Fi24‘
2 Following Atlas ((A4×O8 (2):3):2) is the normalizer of 2A . the element of class 2A is given by a, we can find the other element of class 2A inside the centralizer of of a. The generators for the centralizer of a are given below.
Searching normalizer of ((A4×O8 (2):3):2) inside the above calculated centralizer.The words for the partial normalizers are given by.
The generators for 2A2 are given by
Combining the above calculated partial normalizers we get It just remains to calculate the normalizer of 2A2, This can the required normalizer of ((A4×O8 (2):3):2). The generators be done by constructing the partial normalizers of ((A4×O8 of ((A4×O8 (2):3):2) are given by k3, k5 and k7. (2):3):2) inside the involution centralizers of a and b1. The generators for the involution centralizers of a are given below 4 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
Searching normalizer of ((A4×O8 (2):3):2) inside the electrons. In [43], authors provided a pedagogical above calculated centralizer.The words for the partial overview of the basic challenges of strong correlation, normalizers are given by. 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
Similary we can found the centralizer of b1 using the has strong relationship with quantum mechanics [45]. The method given in [41]. general quantum-mechanical perturbation theory on the nonlinear optical effect in crystals and gives a systematic presentation of the basic concepts and calculation Words for maximal Subgroups of Fi24‘ 1499
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