On Commuting Automorphisms of p-Groups

Abstract

Let G be a group. If the set 𝒜(G) = {α ∈Aut(G) | xα(x) = α(x)x, for all x ∈ G} forms a subgroup of Aut(G), then G is called 𝒜(G)-group. We show that the minimum order of a non-𝒜(G) p-group is p ⁵ for any prime p. We also find the smallest group order of a non-𝒜(G) group. This is related to a question introduced by Deaconescu, Silberberg, and Walls [4]. Moreover, we prove that for any prime p and for all integer n ≥ 5, there exists a non-𝒜(G) group of order p ⁿ .

Key Words:

• Commuting automorphism
• Extraspecial p-group
• Generalized extraspecial p-group
• p-group

2010 Mathematics Subject Classification:

• 20F28

ACKNOWLEDGMENTS

The authors thank Professor S. K. Sehgal, the editor of the Communications in Algebra, and the referee who have patiently read and verified this note.

Notes

Communicated by S. Sehgal.