Commuting_automorphisms_of_auto-Engel_and_auto-Bell_groups

Commuting_automorphisms_of_auto-Engel_and_auto-Bell_groups...

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Unformatted text preview: Extended Abstracts of the 42 nd Annual Iranian Mathematics Conference 5-8 September 2011, Vali-e-Asr University of Rafsanjan, Iran, pp 147-150 COMMUTING AUTOMORPHISMS OF AUTO-ENGEL AND AUTO-BELL GROUPS MOHAMMAD REZA R. MOGHADDAM 1 , HESAM SAFA 2 AND AZAM K. MOUSAVI 3 * Abstract. Let x be an element of a group G , α ∈ Aut ( G ), n ∈ N and A ( G ) = { α ∈ Aut ( G ) | xα ( x ) = α ( x ) x, ∀ x ∈ G } . The autocommutator [ x, n α ] is defined inductively by [ x,α ] = x- 1 x α = x- 1 α ( x ) and [ x, n +1 α ] = [[ x, n α ] ,α ]. The group G is said to be n-auto-Engel, if [ x, n α ] = [ α, n x ] = 1, for all x ∈ G and every α ∈ Aut ( G ), where [ α,x ] = [ x,α ]- 1 . Also, the group G is called an n-auto-Bell group when [ x n ,α ] = [ x,α n ], for every x ∈ G and every α ∈ Aut ( G ). In this paper, we investigate the properties of such groups and their commuting automorphisms set A ( G ). 1. Introduction and Preliminaries Let x 1 and x 2 be the elements of a group G and ϕ x 2 be the inner automor- phism of G defined by x 2 . Then [ x 1 ,x 2 ] = x- 1 1 x x 2 1 = x- 1 1 ϕ x 2 ( x 1 ) denote the commutator of x 1 and x 2 . As in Hegarty [3], if α ∈ Aut ( G ) and x ∈ G then the autocommutator of x and α is defined to be [ x,α ] = x- 1 x α = x- 1 α ( x ). Consider the following definition of the derived subgroup and the centre of G, G = h [ x,ϕ g ] | x,g ∈ G,ϕ g ∈ Inn ( G ) i , Z ( G ) = { x ∈ G | [ x,ϕ g ] = 1 , ∀ ϕ g ∈ Inn ( G ) } . Then one may define, in an analogous manner the following subgroups of G, K ( G ) = h [ x,α ] | x ∈ G,α ∈ Aut ( G ) i , 2000 Mathematics Subject Classification. Primary 20D45; Secondary 20F12, 20E36....
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