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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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