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On the fixed-point set of an automorphism of a group

B. A. F. Wehrfritz

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Let $\phi$ be an automorphism of a group $G$. Under various finiteness or solubility hypotheses, for example under polycyclicity, the commutator subgroup $[G, \phi]$ has finite index in $G$ if the fixed-point set $C_{G}(\phi)$ of $\phi$ in $G$ is finite, but not conversely, even for polycyclic groups $G$. Here we consider a stronger, yet natural, notion of what it means for $[G, \phi]$ to have 'finite index' in $G$ and show that in many situations, including $G$ polycyclic, it is equivalent to $C_{G}(\phi)$ being finite.

Article information

Publ. Mat., Volume 57, Number 1 (2013), 139-153.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F16: Solvable groups, supersolvable groups [See also 20D10] 20E36: Automorphisms of infinite groups [For automorphisms of finite groups, see 20D45]

Automorphism fixed-point set soluble group


Wehrfritz, B. A. F. On the fixed-point set of an automorphism of a group. Publ. Mat. 57 (2013), no. 1, 139--153.

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