ON THE NORMALIZER OF A GROUP IN THE CAYLEY REPRESENTATION

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1 Internat. J. Math. & Math. Sci. VOL. 12 NO. 3 (1989) ON THE NORMALIZER OF A GROUP IN THE CAYLEY REPRESENTATION SURINDER K. SEHGAL Department of Mathematics The Ohio State University Columbus, Ohio (Received October 21, 1987) ABSTRACT If G is embedded as a proper subgroup of X in the Cayley representation of G, then the problem of "if Nx(G) is always larger than G" is studied in this paper. KEY WORDS AND PHRASES: Cayley representation, Wreath product, permutation group A.M.S. SUBJECT CLASSIFICATION CODE. 20B35, 20D25, Let R be the Cayley representation (i.e., the right regular representation) of a group G given by Rill) (z) forall g E 17 and a: E 17. Under the mapping R,thegroup 17 is embedded into a subgroup R(17) of the symmetric group Sta, the group of permutations on the set fl consisting of the elements of the group 17. We identify R(17) with 17 and say that 17 is a subgroup of The centralizer of 17 in Sn consists precisely of the elements of the form (gz). (See Lemma 1.) In particular, if 17 is abelian then 17 is self centralizing in Sn. Also, the normalizer of 17 in Sn is equal to 17- Aut(17) where Aut(17) is the full automorphism group of 17 (see Lemma 2). Suppose that the group 17 is nonabelian. If X is a subgroup of Sn, containing a permutation of the type (gffiffi) for sme g E G Z(G) such that the property G ; x _< Sn (,) holds, then it follows that Nx(G) contains G properly. However, it is easy to see that any element of Sn which normalizes (7 is not always a permutation of the form (a). When the group 17 is abelian, the permutations {z) all lie in 17 and so 17 is self centralizing in Sn. In this way one cannot find a group X satisfying (*) by the above method. However, P. Bhattacharya 1] proved that if 17 is any finite, abelian p group satisfying (*) then Nx(17) > 17. P. Bhattacharya and N. Mukherjjee [2] also prove that if 17 is any finite, nilpotent, Hail subgroup of X satisfying (*) and the Sylow p subgroups of 17 are regular for all primes p dividing the order of 17, then Nx(17) 17. In other words, that X must contain an element of the outer automorphism group of 17.

2 460 S.K. SEHGAL In this paper we will igrove that if 17 is any abelian Hall subgroup of X, satisfying the condition (*) then 17 : Nx(17). We will also give an example to show that the condition of being Hall subgroup is necessary in the above theorem. We will also show that if 17 is any nilpotent, Hall subgroup of X satisfying the condition (*) and the Sylow p subgroups P of 17 do not have a Nx(17). In particular it follows that if 17 is any finite p-group and does not have a factor group isormophic to 2, then 17 < Nx(G) [i.e., the condition being a Hall subgroup is not necessary]. As a corollary it also follows that if 17 is any regular p-group satisfying the conition (*) then G Nx(G). We will give an example to show that the condition of 17 having no factor group isomorphic to 2, is necessary. factor group that is isomorphic to the Wreath product of g, gp then 17 : Lemma 1. Let R be the fight regular representation of a finite group 17 and L, the left regular representation of 17. Under the mappings L and R, the groups L(G) and R(17) are subgroups of S and Csa(R(G)) L(G). Proof: Let (,} e R(G), {h:} e L(G) Hence L(G) C_ Now suppose 1) C8a(R(G)). xg hx xg xg hxg hx hxg hx xg Since Hence Hence Lem Pf: t (zg) z g Hence Aut(G) C. Nsa(R(G)). Conversely, let (ffi,} be an arbitrary element of Nsa(G). Let a 1 So (ffin-l} E R(G). Let 0 (ffiffi,)(ffi,-r}. So 0 sends to 1. Now {ffiffi0} -1 {ffi)(ffi) E R(G). So (ffi:} (ffig)((ffi;o) ((figs0) (;o} since it lies in R(G), i.e., (x.g)0 x,o. g*. Plug in x 1,weget g* go = (xg)o xo.go =, 0 is an autormophism of G = Nsa(R(G)) R(G). Aut(G). Lemma 3: Let 17 be any finite group satisfying the condition (*). Then for any t f (i) G Cl Xa {e}. (ii) X= a.x, (iii) Xa is corn free, i.e., it does not contain any non-identify normal subgroup of X. Proof: Recall that here G is identified with R(G) in G <_ X <_ Sty. Since R is the fight regular representation of G, so R(g) does not fix any a t l except when g e. So G CI Xa

3 NORMALIZER OF A GROUP IN THE CAYLEY REPRESENTATION 461 {e}. Also X acts transitively on fl, [w x" I=1 I=1 el. Now [X.X.] =1 c i=1 el. So X dr. Xa. For part (iii)suppose N X and N C_ X,. So N C t"lexz-lxaz, i.e., if n is an arbitrary element of N, then n can be written as n z-luz for all a: E X and some u E Xa. Here u depends on z, i.e., z-n u-z or tx a tx since u fixes a, i.e., n fixes t for all z X, but X acts transitively on fl = n fixes every element of X = n e = N {e}. Lemma4: (Core Theorem): Let H beany subgroupof dr with [dr" H] n. then dr/eoreh is isomorphic to a subgroup of,.q, where core H is the largest normal subgroup of dr which is contained in H. Proof: Let fl be the set of distinct right cosets of H in dr, i.e., {Hgl Hg2, Hgn} Then the mapping a defined by a(.q) Hg, \ Hgig permutation representation of dr of degree n with Kernel of tr core H. is a transitive Theorem 5: Let dr be a finite abelian, Hall subgroup of X, satisfying the condition (*). Then Nx(dr) >, dr. Proof: Suppose the result is false, i.e., there exists a subgroup X of S satisfying dr X _< S and Nx(dr) dr. Amongst all subgroups of Sn containing dr property, pick X to be smallest. In other words, dr is a maximal subgroup of X Let Idrl P,P2 2 ",P* with p distinct primes. Let Pi be Sylow pi subgroups of dr for i 1,2,...,t. Since dr is a maximal subgroup of X, so Nx(Pi) dr or Nx(Pi) X. Renumber the pi s if necessasry and say Nx(Pi) dr for i l,...,t and Nx(Pi) X for i t+l,...,t. For i l,...,l, Nx(Pi) C,x(Pi) dr. So by Burnside Lemma X has a normal Pi complement. For 3" l + I,...,, Pj,a X =. Cx(Pj),a X dr C_. C,x(Pj) = Cx(PI) X Nx(P). So X has a normal P/ complement Mi for all i = Xa 1 Mi, Xa S which is a contradiction to i=i Lemma 3. In the case where dr is abelian, but not Hall subgroup of X, the result is not true as illustrated by the following example. Example6: Let X Z3 x $3 (a) x (b, clb 3 c 2 1, c-lbc b-). - Let G Z3 Z2 (a) (c) Z6. Let H be the subgroup of X of order 3 generated by the ordered pair (a, b). Then H is not normal in X since (e, e) does not normalize H. So H is core free, of index 6 in X. By Lemma 4, dr X < $6. Now dr is abelian, not Hall subgroup of X and Nx(dr) dr. Theorem 7: Let dr be a finite, nilpotent, Hall subgroup of X, satisfying the condition (*). Suppose that the Sylow p subgroups P of dr do not have a factor group isomorphic to the Wreath product of Z, Zp for all primes p dividing the order of dr. Then Nx(dr) dr. Proof: Suppose the result is false, i.e., there exists a subgroup X of Sa satisfying dr X < St and Nx(dr) dr. Amongst all subgroups of Sfa containing dr properly, pick X to be ix2 smallest. In other words dr is a maximal subgroup of X. Let dr [= 1o "P2 P, here Pi are all distinct primes. Since dr is nilpotent, so dr P x P2 x x Pt where Pi are Sylow Pi subgroups of dr. So we have either Nx(Pi) dr or Nx(Pi) X. Renumber the Pi s if necessary and say Nx(Pi) dr for i 1,...l and Nx(Pi) X for i l + 1,...,t. Let us look at the case i 1,..., l. We have N Nx(Pi) dr. By Yoshida s transfer theorme [3], X has normal Pi complement Mi. Let M I li/=l M So 10i IMI for i 1,...,l. Now for j t+ 1,...,t, Nx(Pj) X. So P, X which implies that Ux(Pj) X and PUx(P) X and dr C_ Pj. Ux(P) = PjCX(Pj) X.

4 462 S.K. SEHGAL For x fl, bylemma3 X G.X=; G X= I; ([ G [,[ X= D x v+,...v IX, Cx(Pj) n M X. c CM(Pj) for j t+ t. M l= -"+ X= Xu is a chactedsfic sub.up of MAG XuAG, which is a concfion to As an immediate corollary to the theorem, we get the result of P. Bhattacharya and N. Mukhcrjee [2]. Corollary $: Let G be a finite, regular 10 subgroup of X and satisfies the condition (*), then " Proof: If G is not a Hall subgroup of X then G is propertly contained in a Sylow p subgroup of X and so Nx(G) >4- So we can assume that G is a Hall subgroup of X. Now G being a regular p group = G does not have a factor group isomorphic to p g,. So Theorem 7 proves the result. Corollary 9. Let G be a finite, nilpotent, Hall subgroup of X, satisfying the condition (*). Suppose further that Sylow p subgroups of G are regular for all primes p dividing the order of G then Nx(G) >+ G. Corollary 10: Let G be a finite p group, satisfying the condition (*). Suppose that G does not have a factor group isomorphic to Zp Z,, then G Nx(G). The condition that the Sylow p subgroups of G in Theorem 6 have the property that it has no homornorphic isomorphic to Z n Zp is necessary. Se example below. Example: Let X be the simple group of order 168. Let G E Sld2(X). Then G " Z2 Z2 so G is nilpotent, Hall subgroup of X. Since H the normalizer of a Sylow 7 subgroup has index 8, so by Lemma 4, G X i.e., G satisfies the condition (*) but Nx(G) G.,.qs, References 1. P. Bhattacharya, On the normalizer of a group in the Cayley representation, Bull. Austrian Math. Soc. 2:5 (1982) P. Bhattacharya and N. Mukherjee, On the normalizer of a subgroup of a finite group and the Cayley embedding; J. of Pure and Applied Algebra 33 (1984) T. Yoshida, Character theoretic transfer; J. of Algebra :52, (1978) B. Huppert, Endliche Gruppen (Springer, New York, 1967). :5. M. ISSACS, Character Theory of Finite Groups. (Academic Press 1976). 6. H. Zassenhaus, The Theory of Groups (Chelsea, 19:56).

5 Journal of Applied Mathematics and Decision Sciences Special Issue on Intelligent Computational Methods for Financial Engineering Call for Papers As a multidisciplinary field, financial engineering is becoming increasingly important in today s economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management. For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy. Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made. Until now, computational methods and models are central to the analysis of economic and financial decisions. However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models. In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches. For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset. Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results. The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making. In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation Authors should follow the Journal of Applied Mathematics and Decision Sciences manuscript format described at the journal site Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at according to the following timetable: Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009 Guest Editors Lean Yu, AcademyofMathematicsandSystemsScience, Chinese Academy of Sciences, Beijing , China; Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong; yulean@amss.ac.cn Shouyang Wang, AcademyofMathematicsandSystems Science, Chinese Academy of Sciences, Beijing , China; sywang@amss.ac.cn K. K. Lai, Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong; mskklai@cityu.edu.hk Hindawi Publishing Corporation