On the exponent of a finite group admitting a fixed-point-free four-group of automorphisms

2011/01/27 by Emanuela Romano, Romano, E., Pavel Shumyatsky +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.1101.5367

openalex publication_date 2011/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A be a group isomorphic with either S₄, the symmetric group on four symbols, or D₈, the dihedral group of order 8. Let V be a normal four-subgroup of A and α an involution in A∖ V. Suppose that A acts on a finite group G in such a manner that C_G(V)=1 and C_G(α) has exponent e. We show that if A≅ S₄ then the exponent of G is e-bounded and if A≅ D₈ then the exponent of the derived group G' is e-bounded. This work was motivated by recent results on the exponent of a finite group admitting an action by a Frobenius group of automorphisms.

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