On the rank of a finite group of odd order with an involutory automorphism
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On the rank of a finite group of odd order with an involutory automorphism Cristina Acciarri1
· Pavel Shumyatsky1
Received: 8 June 2020 / Accepted: 3 November 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract Let G be a finite group of odd order admitting an involutory automorphism φ, and let G −φ be the set of elements of G transformed by φ into their inverses. Note that [G, φ] is precisely the subgroup generated by G −φ . Suppose that each subgroup generated by a subset of G −φ can be generated by at most r elements. We show that the rank of [G, φ] is r -bounded. Keywords Finite groups · Automorphisms · Rank of a group Mathematics Subject Classification 20D45
1 Introduction Let G be a finite group of odd order admitting an involutory automorphism φ. Here the term “involutory automorphism” means an automorphism φ such that φ 2 = 1. We let G −φ stand for the set {g ∈ G | g φ = g −1 } and G φ for the centralizer of φ, that is, the subgroup of fixed points of φ. As usual we denote by [G, φ] the subgroup generated by all elements of G that can be written as g −1 g φ for a suitable g ∈ G. It is well known that [G, φ] is normal in G and φ induces the trivial automorphism of G/[G, φ]. Observe that [G, φ] is precisely the subgroup generated by G −φ . This is because an automorphism of order at most two of a group of odd order is nontrivial
Communicated by John S. Wilson. This research was supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and Fundação de Apoio à Pesquisa do Distrito Federal (FAPDF), Brazil.
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Cristina Acciarri [email protected] Pavel Shumyatsky [email protected]
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Department of Mathematics, University of Brasilia, Brasília, DF 70910-900, Brazil
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C. Acciarri, P. Shumyatsky
if and only if G −φ = {1} (cf Lemma 1(i) in the next section).The following theorem was proved in [10, Theorem B]. Theorem 1 Let G a finite group of odd order admitting an involutory automorphism φ such that the rank of G φ is at most r . Then the rank of [G, φ] is r -bounded. Recall that a rank of a finite group G is the least number r such that each subgroup of G can be generated by at most r elements. Throughout this manuscript we use the term “(a, b, c . . . )-bounded” to mean “bounded from above by some function depending only on the parameters a, b, c . . . ”. Since in a finite group of odd order with an involutory automorphism φ there is a kind of (very vague) duality between G φ and G −φ , in this paper we address the question whether a rank condition imposed on the set G −φ has an impact on the structure of G. We emphasize that G −φ in general is not a subgroup of G and therefore the usual concept of rank does not apply to G −φ . Instead we impose the condition that each subgroup of G generated by a subset of G −φ can be generated by at most r elements. Our main result is as follows. Theorem 2 Let G be a group of odd order admitting an involutory automorphism φ and suppose that any subgroup generated by a subset of G −φ can be generated by r elements. Then [G
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