Skew-morphisms of cyclic $p$-groups

TL;DR

This paper classifies all skew-morphisms of cyclic p-groups for odd primes, expanding understanding of group factorizations and automorphisms in algebraic structures.

Contribution

It provides a complete classification of skew-morphisms of cyclic p-groups for odd primes, a previously uncharacterized class of algebraic mappings.

Findings

Complete classification of skew-morphisms for cyclic p-groups

Characterization of associated group factorizations

Insights into automorphism structures of cyclic groups

Abstract

Let $G$ be a finite group having a factorisation $G=AB$ into subgroups $A$ and $B$ with $B$ cyclic and $A\cap B=1,$ and let $b$ be a generator of $B$. The associated skew-morphism is the bijective mapping $f:A \to A$ well defined by the equality $baB=f(a)B$ where $a\in A$. In this paper, we shall classify all skew-morphisms of cyclic $p$-groups where $p$ is an odd prime.