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Forcing a finite group to be Abelian. (English) Zbl 1275.20019
From the introduction: Conditions ensuring that a finite group $$G$$ is Abelian are given in terms of element centralisers and automorphisms of $$G$$.
H. J. Zassenhaus [Proc. Glasg. Math. Assoc. 1, 53-63 (1952; Zbl 0049.16002)] discovered an elegant theorem: $$G$$ is Abelian if and only if $$N_G(A)=C_G(A)$$ for all Abelian subgroups $$A$$ of $$G$$.
Theorem 1.1. If $$G$$ is a finite nonabelian group, then there exists some element $$x\in G\setminus Z(G)$$ such that $$C_G(C_G(x))<N_G(C_G(x))$$.
As a direct consequence we have the following Zassenhaus type result: Corollary 1.1. If $$G$$ is a finite group, then $$G$$ is Abelian if and only if $$N_G(C_G(x))=C_G(C_G(x))$$ for all elements $$x\in G\setminus Z(G)$$.
Let $$I=\text{Inn}(G)$$ denote the group of inner automorphisms of $$G$$ and let $$J=J(G)$$ denote the group of the so-called class-preserving automorphisms of $$G$$. Here $$\tau\in J$$ provided that for every $$x\in G$$ there exists some $$g\in G$$ (depending on $$x$$) such that $$x^\tau=x^g$$.
Theorem 1.2. Let $$G$$ be a finite group. Then $$G$$ is Abelian if and only if $$\text{Inn}(G)\leq\Phi(J(G))$$.
Theorem 1.3. Let $$G$$ be a finite group. Then every Abelian subgroup of $$G$$ is the fixed point subgroup of some coprime automorphism of $$G$$ if and only if $$G$$ is an Abelian group of odd square-free exponent.
##### MSC:
 20D45 Automorphisms of abstract finite groups 20D25 Special subgroups (Frattini, Fitting, etc.) 20E34 General structure theorems for groups
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