Coprime automorphisms of finite groups
Cristina Acciarri, Robert M. Guralnick, Pavel Shumyatsky

TL;DR
This paper investigates how coprime automorphisms influence the structure of finite groups, establishing bounds on the rank of certain subgroups and deriving properties like solubility and nilpotency under specific conditions.
Contribution
It introduces bounds on the rank of the subgroup generated by commutators related to coprime automorphisms and proves new results on the structure and properties of these subgroups.
Findings
The rank of [G,α] is bounded by (e, r).
If all elements of I_G(α) have odd order, then [G,α] has odd order.
Pairs of elements generating soluble or nilpotent subgroups imply [G,α] is soluble or nilpotent.
Abstract
Let be a finite group admitting a coprime automorphism of order . Denote by the set of commutators , where , and by the subgroup generated by . We study the impact of on the structure of . Suppose that each subgroup generated by a subset of can be generated by at most elements. We show that the rank of is -bounded. Along the way, we establish several results of independent interest. In particular, we prove that if every element of has odd order, then has odd order too. Further, if every pair of elements from generates a soluble, or nilpotent, subgroup, then is soluble, or respectively nilpotent.
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