A fusion result of Glauberman has as a consequence the fact that a finite group admitting a fixed-point-free automorphism has a normal Sylow 2-subgroup (and in particular, is solvable) if all ...nontrivial fixed-point subgroups have odd order.
Let $G$ be a finite group, $\sigma$ an automorphism of $G, M$ a $\sigma$-invariant subgroup of $G$, and $n$ a fixed integer. If $\sigma (g) \in g^nM$ for all $g \in G$ then there exists a ...$\sigma$-invariant normal subgroup $K$ of $G$, contained in $M$, with $\sigma (g) \in g^nK$ for all $g \in G$.
Let G=Fq⋊〈β〉 be the semidirect product of the additive group of the field of q=pn elements and the cyclic group of order d generated by the invertible linear transformation β defined by ...multiplication by a power of a primitive root of Fq. We find an arithmetic condition on d so that every endomorphism of G is determined by its values on (1,1) and (0,β). When that is the case, we determine the fixed point free automorphisms of G. If d equals the odd part of q−1 then we count the fixed point free automorphisms of G—such exist only when p is a Fermat prime.
Let
G be a group and
A
a group of automorphisms of
G. An
A
-orbit of
G is a set of the form
{
g
α
|
α
∈
A
}
, where
g is an element of
G. The aim of this paper is to prove that if
A
is abelian and
G ...is a union of a finite number of
A
-orbits then
G admits a normal abelian subgroup of finite index. This result answers affirmatively a question raised by Neumann and Rowley (1998) in
4.
.
We prove that a finite group having a fixed-point-free automorphism in the Fitting subgroup of its automorphism group must be abelian of rather restricted structure. As a consequence, no finite ...nonabelian group could have a fixed-point-free automorphism in the Frattini subgroup of its automorphism group.
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