Short communication
The Combinatorial Isomer Enumeration of 1, 3, 5-trimethylbenzene by Fujita's Topological Index
Ali Moghani,1* Mohammad Reza Sorouhesh2 and Soroor Naghdi1
1 Department of Color Physics, Institute for Colorants Paints and Coatings, Tehran, Iran 2 Department of Mathematics, Islamic Azad University, South Tehran branch, Tehran, Iran * Corresponding author: E-mail: moghani @icrc.ac.ir
Received: 24-08-2009
Abstract
The unmatured full non-rigid group of 1,3,5-trimethylbenzene is isomorphic to the wreath product of the cyclic group of order three and the symmetric group of order six on three letters (see Int. J. Quantum Chem. 2007, 107, 340 and Bull. Chem. Soc. Jpn. 2008, 81, 279). In this paper the unit subduced cycle index table introduced by S. Fujita of the full nonrigid group 1,3,5-trimethylbenzene of order 162 are successfully derived for the first time.
Keywords: Non-rigid molecule group, Unit subduced cycle index, 1,3,5-trimethylbenzene.
1. Introduction
Chemically, for any non-rigid molecules, there are one or more contortional large amplitude vibrations, such as inversion or internal rotation, which lead to tunneling splitting. Because of this deformability, such molecules exhibit some remarkable properties of intermolecular dynamics, which can be surveyed more easily by resorting to Group theory. A complete set of molecular conversion operations that commute with the nuclear motion operator includes total rotation operations, describing the molecule rotating as a whole and the non-rigid tunneling motion operations that depict molecular moieties moving with respect to the rest of the molecule. Such a set forms a group called the full non-rigid group (f-NRG).1-3
In 1960's, Longuet-Higgins3 investigated the symmetry groups of non-rigid molecules, where changes from one conformation to another can occur effortlessly. The method described here is appropriate for molecules consisting of a number of CH3 groups attached to a rigid framework.3-10
Through the present study, we intend to probe the unite subduced cycle indices for the f-NRG 1,3,5-tri-
methylbenzene as are presented introduced by S. Fujita.11-18
The motivations for this study, utilizing GAP19 are the authors previous works on the chemical molecules.20-25
2. Experimental
In this section, in respect to Fujita's symbols, we describe some notations that will be kept thoroughly in this paper.
Let G be an arbitrary finite group and h1, h2, e G, we say h1 and h2 are Q-conjugate, denoted by h2 - Qh2, if we can find any t e G such that t1 (h1) t = (h2). Obviously, this Q-conjugacy is an equivalence relation on group G and it generates equivalence classes that are called dominant classes. Therefore G is partitioned into the dominant classes as follows: G = K1 + K2+ ••• + Ks. Now assume that an action P of G on a set X and a subgroup H of G are given. So by considering the set X consisting of all the H's, right cosets,and the partition of G induced by these cosets; G = ®rfHgi, we have an action of G on X and a permutation representation signified by G(/H) correspon-
dingly. If Gj and Gj be any subgroups of an arbitrary finite group G, a subduced representation denoted G(/Gi) I Gj is known as a subgroup of the coset representation G(/Gi) that contains only the elements associated with the elements in Gj.11-14
The table of marks of a finite group G is a square matrix M(G) = (mik) 11 < k<rrwhere mikis the number of right cosets of Gk in G which remain fixed under right multiplication by the elements of Gi.26
Let Gi and Gj be two subgroups of an arbitrary finite group G. A unit subduced cycle index (USCI) is delinea-
ted as Z(G(/Gi) I Gj,sd) = ngeQs(jg where Q is a transversal set for the double coset decompositions concerning Gi and Gj for i,j = 1,2,3, •••, s and sdgjGJ/g^g n Gj|, the reader is recommended to consult the ref. 11-18 and 24.
If K and H are groups and H acts on the set r, then the wreath product of K by H, denoted by Kwr H is identified with Kr : H is semi direct product Kr by H such that Kr = {f | f: r ^ K}.8-10, 25
Suppose that H is a cyclic subgroup of order n of a finite group G. Then the maturity discriminant of H, denoted by m(H), is an integer number delineated by |NG(H) :
Table 1: The unit subduced cycle indices for the f-NRG of TMB (i.e. 2(0(10)10, sd)) fo i, j = 1 to 34
USCI	G1	G2	G3	G4	G5	G6	G7	G8	G9	G10	G11	G12	G13	G14	G15	G16	G17
G(/G1)	sf	s?81	s54 3	s54 3	s54 3	s54 3	s54 3	s54 3	s6?7	s?7 6	s6?7	s?7 6	s?7 6	s6?7	s98	ss198	s98
G(/G2)	S81	s36s9	s3?7	s?7 s3	s ?7 s3	s?7 s3	s?7 s3	s?7 s3	s 9s 9 s6s3	s^3	s^3	s16?s33	s3?s3	s 9s 9 s6 s3	s9 s9	s9	s9
G(/G3)	sf	s?7	s54	s18 3	s18 3	s18 3	s18 3	s^8	s9	s9	s9	s9	s?7 ?	s69	s18 3	s18 3	s9
G(/G4)	s?	s?7 b?	s|8	s1?s18 31	s18 3	s18 3	s18 3	s|8	s9 6	s6s9 9?	s9 6	s9 6	s9 6	s9 6	s6 9	s4s6 93	s6 9
G(/G5)	s?	s?7 s?	s|8	s18 s3	s1?s18 31	s18 s3	s18 s3	s|8	s6s9 6?	s9	s9	s9 s6	s9	s69	s18	s6 s9	s4s6 91
G(/G6)	sf	s?7 ?	s^8	s18 3	s18 3	s1?s18 31	s18 3	s^8	s9	s9	s9	s6s9 9?	s9	s69	s6	s4s6 93	s4s6 93
G(/G7)	s?	s?7 ?	s|8	s18 3	s18 3	s18 3	s 1?s 18 31	s18 3	s9	s9 6	s6s9 6?	s9 6	s9 6	s9 6	s6 9	s4s6 93	s?s1 93
G(/G8)	s?	s?7 s?	s18	s18	s18 s3	s18	s318	s16s6	s9 s6	s9 s6	s9 s6	s9 s6	s9 s6	s8s3 6?	s6	s6	s6 s9
G(/G9)	s?7	9 9 s?s1	s?	s9	s6s9 31	s9	s9	s9	39 s6si	33 s6s3	33 s6s3	33 s6s3	33 s6 s3	s9	s9	s3	s?s3 93
G(/G10)	s?7	s1?s3 ?1	s9 3	s6s9 31	s9 3	s9 3	s9 3	s9 3	s 3s3 s6s3	s 3s 3s 3 s6s?s1	s 4s s6s3	s 4s s6s3	s4s1 63	s 3s3 s6s3	s3 9	s?s3 93	s3 9
G(/G11)	s?7	s1?s3 ?1	s9	s9	s9	s9 s3	s6s9 31	s9	s 3s3 s6s3	s 4s s6s3	s 3s 3s 3 s6s?s1	s 4s s6s3	s4s1 63	s 3s3 s6s3	s9	s?s3 93	s s6 93
G(/G12)	s?7	s1?s3 ?1	s9	s9	s9	s6s9 31	s9	s9	33 s6s3	s 4s 63	s4s 63	333 s6s?s1	s4s1 63	33 s6s3	s9	s?s3 93	s?s3 93
G(/G13)	s?7	s1?s3 ?1	s?7	s9 3	s9 3	s9 3	s9	s9 3	s 3s3 s6s3	s 4s s6s3	s 4s s6s3	s 4s s6s3	s1?s3 ?1	s 3s3 s6s3	s9 3	s9 3	s3 9
G(/G14)	s?7	9 9 s?s1	s9	s9	s9	s9	s9	s8s3 31	s9	s 3s 3 s6s3	s 3s 3 s6s3	s 3s 3 s6s3	s 3s 3 s6s3	s6s36s13	s9	s9	s9
G(/G15)	si8	s?9	si8	s6	si8	s6	s9	s6	s?9	s3	s3	s3	s?9	s63	s;8	s6	s6
G(/G16)	si8	s?9	si8	s4s6 31	s6 3	s4s6 31	s 4s 6 31	s6	s3 6	s?s3 6?	s?s3 6?	s?s3 6?	s?9	s63	s6 3	s4s6 31	s6 3
G(/G17)	si8	s?9	s6	s6 s3	s4s6 31	s4s6 31	? 1? s ?s 1? 31	s6	s?s3 6?	s3 s6	s s6 6?	s?s3 6?	s 63	s3	s6	s6	s4s6 31
G(/G18)	si8	s?9	s6	SlS1l 31	s4s6 31	s6	s 4s 6 31	s6	s?s3 6?	s s6 6?	s?s3 6?	s3	s 63	sf	s6	s6	s6
G(/G19)	si8	s9 ?	si8	s6 3	s6 3	s6	s6 3	s4s6 31	s3 6	s3	s3 6	s3	s9 ?	s?s3 6?	s6 3	s6 3	s?
G(/G20)	si8	s?9	s6	s4s6 31	s4s6 31	? 1? s ?s 1? 31	s6	s6	s?s3 6?	s?6s?3	s3	s s6 6?	s 63	s3	s6	s?	s6
G(/G21)	si8	s?9	si8	s6	s6	s6	s6	s6	s3	s3	s3	s3	s?9	s3	s6	s?	s?
G(/G22)	s1	3 3 s?s1	s3	s?s3 31	s?s3 31	s s6 31	s3 3	s3 3	3 s6s13	3 s6s13	s6s3	s s3 s3s?	s6s3	s3 3	s3	s3 3	s3 3
G(/G23)	s1	3 3 s?s1	s9	s3	s9	s3 s3	s3 s3	s3	3 3 s?s1	s6s3	s6s3	s6s3	s 3s 3 s?s1	s3	s9	s3	s3 s3
G(/G24)	s1	33 s?s1	s3	s3	s?s3 31	s?s3 31	s s6 31	s3	s6s13	s6s3	s s3 3?	s6s13	s6s3	s3	s3	s3	s?s3 31
G(/G25)	s1	3 3 s?s1	s3	s s6 31	s?s3 31	s3 3	s ?s 3 31	s3 3	3 s6s13	s s3 s3s?	3 s6s13	s6s3	s6s3	s3 3	s3 3	s3 3	s3 3
G(/G26)	s1	s 4s s?s1	s9	s?s3 31	s3	s?s3 31	s ?s 3 31	s3 s3	s6s3	s6s?s1	s6s?s1	s6s?s1	s 4s s?s1	s6s3	s3	s?s3 31	s3 s3
G(/G27)	s1	s?s3 31	sj	s3	s3	s3	s3	33 s?s1	s3	s6s3	s6s3	s6s3	33 s?s1	s3?s?s1	s3	s3	s9
G(/G28)	s1	s3 ?	s6	s6	s6	s6	sf	s?	s?3	s3 ?	s3 ?	s3 ?	s3 ?	s6	s6	s6	s6
G(/G29)	s1	s?3	s6	s?	s6	s?	s?	s6	s?3	s6	s6	s6	s?3	s3 s?	s6	s?	s?
G(/G30)	s1	s?3	sj	s?	s6	s?	s?	s?	s?3	s6	s6	s6	s?3	s6	s6	s?	s?
G(/G31)	s1	s3	s3	s3	s3	s3	s3	s3	s3	s3	s3	s3	s3	s3	s3	s3	s3
G(/G32)	s1	s?si	s3	s3	s3	s3	s?	s3	s?s1	s?s1	s?s1	s?s1	s?s1	s3	s3	s3	s3
G(/G33)	s1	s?	sj	s?	sj	s?	s?	s?	s?	s?	s?	s?	s?	s?	sj	s?	s?
G(/G34)	s1	s1	s1	s1	s1	s1	s1	s1	s1	s1	s1	s1	s1	s1	s1	s1	s1
CG (H)| in addition, the dominant class of K n H in NG(H) is the union of t = p (n)/m(H) conjugacy classes of G where p is Euler function, i.e. the maturity of G is clearly defined by examining how a dominant class corresponding to H contains conjugacy classes. If t = p (n)/m(H) = 1, the group G should be matured, but if m(H) < p (n) or t > 2, the group G is an unmatured concerning subgroup H.13-17 Theorem: The wreath products of the matured finite groups are again a matured group, but the wreath products of some finite groups with at least one unmatured group should be an unmatured group.25
3. Results and Discussion
According to the above Theorem and ref. 27, the f-NRG 1,3,5-trimethylbenzene (TMB) is an unmatured group isomorphic to C3 wr S3 of order 162, see Figure 1.
In order to compute the mark table M34x34 and then the set SSGX of TMB with the symmetry X = C3 wr S3, run the program below in the GAP prompt as follows:
LogTo("1,3,5-trimethylbenzene.txt"); c3:=CyclicGroup(IsPermGroup,(3));
Table 1 (continued)
USCI	G18	G19	G20	G21	G22	G23	G24	G25	G26	G27	G28	G29	G30	G31	G32	G33	G34
G(/G1)	s18 »9	s18 9	s18 9	s18 9	s18 9	s18 9	s18 9	s18 9	s18 9	s18 9	s6 27	s6 27	s6 27	s3 54	s3 54	s2 81	si62
G(/G2)	S9 s9	s9 s9	s9	s9 s9	s 3s3 s18s9	s 3s3 s18s9	s 3s3 s18s9	s 3s3 s18s9	4 s18s9	s 3s3 s18s9	s3 27	s3 27	s3 27	s3 27	s54s27	s81	s81
G(/G3)	S18 3	s18 3	s9	s18 3	s3 18	s 9	s3 18	s3 18	s 9	s 9	s9	s9	s9	s3 18	s3 18	s2 27	s54
G(/G4)	S6 »9	s6 9	s4s6 93	s6 9	s2s3 18 6	s3 18	s3 18	s s6 18 6	s2s3 18 6	s3 18	s9	s2 27	s2 27	s54	s18	s2 27	s54
G(/G;)	s9	s9	s4s6 93	s9	s2s3 18 6	s9 s6	s2s3 18 6	s2s3 18 6	s3 18	s3 18	s9	s6 s9	s6 s9	s3 18	s18	s2 27	s54
G(/Gj)	s9	s9	s2s12 93	s9	s s6 18 6	s3 18	s2s3 18 6	s3 18	s2s3 18 6	s3 18	s9	s2 27	s2 27	s54	s18	s2 27	s54
G(/G7)	s6 9	s6 9	s6 9	s9 6	s3 18	s3 18	s s6 18 6	s2s3 18 6	s2s3 18 6	s3 18	s6 9	s2 27	s2 27	s54	s18	s2 27	s54
G(/G„)	s4s6 93	s4s6 93	s6 s9	s9	s3 18	s3 18	s3 18	s3 18	s3 18	s2s3 96	s2 27	s9	s2 27	s3 18	s54	s2 27	s54
G(/G,)	s9	s9	s2s3 93	s9	s s3 18 3	33 s6 s3	s s3 18 3	33 s6 s3	s18s9	s9	s9	s9	s9	s9	s18s9	s27	s27
G(/G10)	s9	s9	s2s3 93	s3 9	s s3 s18s3	s18s9	s18s9	s18s9	s18s6s3	s18s9	s9	s27	s27	s27	s18s9	s27	s27
G(/G11)	s9	s9	s3 s9	s9	s18s9	s18s9	s s3 s18s3	s s3 s18s3	s18s6s3	s18s9	s9	s27	s27	s27	s18s9	s27	s27
G(/G12)	s9	s9	s s6 93	s9	s s3 96	s18s9	s s3 18 3	s18s9	s18s6s3	s18s9	s9	s27	s27	s27	s18s9	s27	s27
G(/G13)	s9 3	s9 3	s3 9	s9 3	s18s9	s3s3 s6s3	s18s9	s18s9	s 4s s6s3	3 3 s6s3	s3 9	s3 9	s3 9	s3 9	s18s9	s27	s27
G(/G14)	2 2 s9s3	s2s3 93	s9	s9	s9	s9	s9	s9	s18s9	2 s9s6s3	s27	s9	s27	s9	s27	s27	s27
G(/G15)	s3	s3	s3	s3	s3	s29	s3	s3	s3	s3	s3	s3	s3	s3	s3	s29	s18
G(/G16)	s3	s3	s6 3	s6 3	s3 6	s3 6	s3 6	s3 6	s2s3 62	s3	s3	s2 9	s2	s18	s3	s29	s18
G(/G17)	S92	S92	s3	s?	s3 s6	s s3 s6s2	s s3 s6s2	s3 s6	s3	si8	s3	s2	s2	s18	s3 s6	s2	s18
G(/G18)	S92	S92	s3	s2	si8	s2s2 s6s3	s4s3 32	s2s2 s6s3	s3	si8	s2	s3	s2	s3	s18	s2	s18
G(/G19)	S4S6 31	s4s6 31	s2 9	s6 3	s2s3 62	s6 3	22 s6s3	s18	s3 6	s2s3 62	s2 9	s6 3	s2 9	s3 6	s18	s2 9	s18
G(/G20)	S92	S92	s4s6 31	s2 s9	si8	22 s6s3	s18	22 s6s3	s3	si8	s3	s2	s2 s9	s18	s3 s6	s2	s18
G(/G21)	s3	s3	s29	s5s3 31	s2s3 62	s3	s3	s18	s3	s3	s29	s29	s s3 93	s18	s18	s2	s18
G(/G22)	s9	s9	s2s3 31	s9	3 s6s13	s3	s9	s6s3	s6s3	s9	s3 3	s3 3	s3 3	s9	s6s3	s9	s9
G(/G23)	s3 s3	s3	s3	s3 s3	s6s3	s3s3 s2si	s6s3	s6s3	s6s3	s3 s3	s3	s3 s3	s3 s3	s3 s3	s6s3	s9	s9
G(/G24)	s9	s9	s3	s9	s6s3	33 s2si	s6s3	s6s3	s6s3	s9	s3	s9	s9	s9	s6s3	s9	s9
G(/G25)	s9	s9	s3 3	s9	s6s3	s6s3	s3 3	3 s6s13	s6s3	s9	s3 3	s9	s9	s9	s6s3	s9	s9
G(/G26)	s3 s3	s3 s3	s3	s3	s6s3	s6s3	s3 s3	s6s3	s6s2s1	s6s3	s3 s3	s9	s9	s9	s6s3	s9	s9
G(/G27)	33 S2S1	33 s2si	s9	s3	s9	s3	s9	s9	s6s3	s32s2s1	s9	s3	s9	s3	s9	s9	s9
G(/G28)	s2	s2	s6	s2 3	s3 2	s3 2	s3 2	s3 2	s3 2	s6	s6	s23	s3 2	s6	s3 2	s6	s6
G(/G29)	s6	s6	s2	s2 s3	s23	s23	s6	s6	s6	s3 s2	s23	s23	s23	s3 s2	s6	s2 s3	s6
G(/G30)	s2	s2	s2	s s3 31	s23	s23	s6	s6	s6	s6	s23	s23	s23	s6	s6	s3	s6
G(/G31)	s3	s3	si	s3	s3	s3	s3	s3	s3	s3	s3	s3	s3	s3	s3	s3	s3
G(/G32)	si	si	s3	s3	s2si	s2si	s2si	s2si	s2si	s3	s?	s3	s3	s3	s2si	s3	s3
G(/G33)	s2	s2	s2	s2	s2	s2	s2	s21	s2	s2	s2	s2	s2	s2	s2	s2	s2
G(/G34)	s1	s1	s1	s1	s1	s1	s1	si	s1	s1	s1	s1	s1	s1	s1	s1	s1
s3:=SymmetricGroup(IsPermGroup,(3)); X:=WreathProduct(c3,s3); Order(X); IsPermGroup(X);
s:=ConjugacyClassesSubgroups(X); Sort("s");
M:=TableOfMarks(X); Display(s); LogTo( ); Print("1,3,5-trimethylbenzene.txt"
Column26:=A*(Inv);
Print("Column26", "\n");
'In");
Afterwards, it can be seen that the non-redundant set of subgroups of X with size 34 consists of the following elements required to calculate all the Fujita's topological indices (i.e. USCIs) as collected in Table 1:
Now, we utilize GAP to calculate all the USCIs of X for TMB. As a matter of fact, for instance, to calculate the column no. 26 in Table 1 (i.e. for G26 of order 18) equiva-lently for i = 1 to 34, Z(G(G) I G26, sd), the following program is operated in GAP system as well:
G26=GroupWithGenerators((7,8,9),(1,2,3)(4,5,6),(1,4) (2,5)(3,6));
M26:=TableOfMarks(G26); Inv:=(M26)" -1; S26:=ConjugacyClassesSubgroups(G26); Sort(s26);
A:=[[18,0,0,0,0,0,0,0,0,0,0,0], [9,9,0,0,0,0,0,0, 0,0,0,0], [6,0,6,0,0,0,0,0,0,0,0,0], [6,0,0,6,0,0,0,0 ,0,0,0,0], [6,0,0,0,6,0,0,0,0,0,0,0], [6,0,0,0,0,6,0, 0,0,0,0,0], [3,3,3,0,0,0,3,0,0,0,0,0], [3,3,0,3,0,0,0 ,3,0,0,0,0], [3,3,0,0,3,0,0,0,3,0,0,0], [3,3,0,0,0,3, 0,0,0,3,0,0], [2,0,2,2,2,2,0,0,0,0,2,0], [1,1,1,1,1,1 ,1,1,1,1,1,1]];
Figure 1: Structure of 1,3,5-trimethylbenzene
4. Conclusions
By applying similar and above calculation for other columns, we are able to calculate Fujita's combinatorial enumeration USCI table of 1,3,5-trimethylbenzene stored in Table 1 which would also be valuable in other applications such as in the context of chemical applications of graph theory and aromatic compounds.1-18' 28
5. Acknowledgment
The authors were in part supported by a grant from Islamic Azad University, South Tehran Branch.
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Povzetek
Nezrela, popolnoma nerigidna grupa 1,3,5-trimetilbenzena je izomorfna s spletnim produktom ciklične grupe tretjega reda in simetrične grupe šestega reda na treh črkah (gl. Int. J. Quantum Chem. 2007, 107, 340 and Bull. Chem. Soc. Jpn. 2008, 81, 279).V tem prispevku prvič uspešno izpeljemo tabelo enotnih zmanjšanih cikličnih indeksov (ki jo je vpeljal S. Fujita) za popolnoma nerigidno grupo 1,3,5-trimetilbenzena reda 162.