Computation of orders and cycle lengths of automorphisms of finite solvable groups

TL;DR

This paper develops algorithms to compute automorphism orders and element cycle lengths in finite solvable groups, providing correctness proofs and analyzing their theoretical complexity.

Contribution

It introduces new algorithms for automorphism order and cycle length computation in finite solvable groups, with proofs and complexity analysis.

Findings

Algorithms are correct and efficient for finite solvable groups.

Complexity analyses of classical algorithms are provided.

Theoretical bounds for algorithm performance are established.

Abstract

Let $G$ be a finite solvable group, given through a refined consistent polycyclic presentation, and $\alpha$ an automorphism of $G$, given through its images of the generators of $G$. In this paper, we discuss algorithms for computing the order of $\alpha$ as well as the cycle length of a given element of $G$ under $\alpha$. We give correctness proofs and discuss the theoretical complexity of these algorithms. Along the way, we carry out detailed complexity analyses of several classical algorithms on finite polycyclic groups.