Hall Skew-morphisms and Hall Cayley maps of finite groups

TL;DR

This work addresses the problem of characterizing finite groups $H$ that admit skew-morphisms whose order is coprime to $|H|$ and provides a comprehensive framework based on Hall factorizations $G=HK$ with $H$ Hall and $K$ cyclic. It develops a structural theory that reduces the classification to the socle $N=T_1\times\cdots\times T_r$ of nonabelian simple factors and an accompanying $(H_1,\dots,H_r)$-decomposition with coprimality constraints, culminating in a full description of Hall skew-morphisms and their automorphism groups (Theorem skewG). Extending to combinatorial symmetry, it classifies vertex-rotary core-free Hall Cayley maps by expressing the ambient group as a product involving almost simple components and describing the underlying coset graphs via (bi-)direct products of component graphs (Theorem maps). The results yield solvable-case explicitly, produce infinite families with arbitrarily many simple factors, and connect skew-morphism theory to highly symmetric Cayley maps and coset-graph constructions, with concrete examples from classical groups and Singer cycles.

Abstract

A characterization is given of finite groups $H$ that have skew-morphisms of order coprime to the order $|H|$, and their skew-morphisms. A complete classification is then given of the automorphism groups and the underlying graphs of vertex-rotary core-free Hall Cayley maps.

Theorems & Definitions (32)

• Definition 1.1
• Theorem 1.3
• Corollary 1.4
• Theorem 1.6
• Corollary 1.7
• Theorem 1.8
• Lemma 2.1
• proof
• Lemma 2.2
• proof