ON THE NORMALIZER OF A GROUP IN THE CAYLEY REPRESENTATION

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.

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 ( g f f i f f i ) 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 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 factor group that is isomorphic to the Wreath product of g, gp then 17 : 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. (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 {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}. In the case where dr is abelian, but not Hall subgroup of X, the result is not true as illustrated by the following example.
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. 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 _ ,.qs, i.e., G satisfies the condition (*) but Nx(G) G.