Central Automorphisms Research Articles

Groups with prescribed automorphism group: A clarification

In Theorems 1 and 2 of [] necessary and sufficient conditions were given for a group G to have a finite automorphism group Aut G and a semisimple subgroup of central automorphisms AutcG. Recently it occurred to us, as a result of conversations with Ursula Webb, that these conditions could be stated in a much simpler and clearer form. Our purpose here is to record this reformulation. For an explanation ofterminology and notation we refer the reader to [1].

On Central Root Automorphisms of Finite Generalized Octagons

Certain techniques, developed in order to study the structure of finite projective planes which admit (non-trivial) elations, are applied to central root automorphisms of generalized octagons.

Picard groups and automorphism groups of integral group rings of metacyclic groups

Let K be a Dedekind domain with quotient field K, and let A be an H order in a separable K-algebra A. A. Frohlich [4] introduced the locally free Picard group of A, which is denoted by LFP(,4). This is defined to be the group of all isomorphism types of invertible /i-/i-bimodules in A which are locally free as left n-modules. As usual, the locally free class group of ,4 is denoted by C(A). It is noted that if/i is commutative, then LFP(/I) z C(/i). Let Aut(/l) be the group of all automorphisms of n and let In(A) be the subgroup of Aut(/i) consisting of all inner automorphisms of/i. Let cent@) denote the center of /i. An automorphism f of /i is called a central automorphism iff(c) = c for any c C ccnt(/i). WC denote by Autccnt(/l) the subgroup of Aut(/l) consisting of all central automorphisms of A, and define Outcent = Autcent(/i)/In(/l). Fundamental properties of the groups LFP(A) and Outcent were systematically studied by Friihlich, Reiner and Ullom (4, 5 1. Let G be a finite group and let LG be its integral group ring. We denote by C, the cyclic group of order n, by D, the dihedral group of order 2n and by H, the quaternion group of order 4n. Frohlich, Reiner and Ullom computed LFP(ZG) and Outccnt(LG) when G = D,, II, or D,, p an odd prime. 286 0021.8693/82/08028~25%02.00/0

Near central automorphisms of abelian torsion groups

Near central automorphisms of abelian torsion groups

Groups in which each element commutes with its endomorphic images

Let G be a finite group in which each element commutes with its endomorphic images. We will give a counterexample to the conjecture that G is abelian. Let G be a finite group in which each element commutes with its endomorphic images. We will give a counterexample to the conjecture that G is abelian. For g and h in G, g and g-1(h-1lgh) commute. Thus G satisfies the identical relation (x, y, x) = 1. Therefore G satisfies the identical relations (x, y, z, w)=1 and (x, y, z)3=I (see [1, p. 322]). Hence G is nilpotent of class <3 and if G is a p-group for a prime ps3, G is nilpotent of class <2. We will exhibit for any prime p a nilpotent class 2 p-group in which each element commutes with all of its endomorphic images. Let p2 G = (a,, a2, a3, a4:a = 1, (ai, aj, ak) = 1, (1 < i,j, k < 4) p p ~~~~~~~~p and (1) (a,, a2) = al, (2) (a,, a3) = a3, (3) (a1, a4) = a4, (4) (a2, a3) = a2, (5) (a2, a4) = 1, (6) (a3, a4) = a3). Let E(G) and A (G) denote the endomorphisms and automorphisms of G respectively. Let EZ(G) = {GCE(G):0(G) <Z(G) } and AZ(G) {OCA(G):O(g)g-1E'Z(G) for all gEEG } =central automorphisms of G. We will prove THEOREM. E(G) =EZ(G)kUAZ(G). COROLLARY. Each element in G commutes with its endomorphic images. LEMMA 1. G| =p PROOF. Clearly from the defining relations I GI <p8. We will construct a group of order p8 which satisfies the defining relations of G. Let Received by the editors February 11, 1970 and, in revised form, April 24, 1970. AMS 1969 subject classifications. Primary 2025, 2040.

The Central Automorphisms of a Finite Group

The Central Automorphisms of a Finite Group

Central Automorphisms of a Finite p-Group

Central Automorphisms of a Finite p-Group

Central automorphisms of a finite𝑝-group

Central automorphisms of a finite𝑝-group

The Structure and Order of the Group of Central Automorphisms of a Finite Group*

The Structure and Order of the Group of Central Automorphisms of a Finite Group^*