Regular automorphisms of periodic $p'$-groups need not be splitting

Abstract

For a fixed prime $p$, a group is called a $p'$-group if the orders of all its elements are coprime to $p$. It is well known that in a finite $p'$-group an automorphism of order $p$ is regular if and only if it is a splitting automorphism of period $p$. In the Kourovka Notebook, V.\,D.~Mazurov asked whether these two notions coincide for periodic $p'$-groups. We answer this question in the negative by constructing a periodic group of prime period $q$ that admits a regular automorphism of order $p \neq q$ which is not splitting.