Computing the Table of Marks of a Cyclic Extension

TL;DR

This paper introduces an algorithm to compute the subgroup pattern, specifically the table of marks, of cyclic extensions and solvable groups based on their subgroups' patterns.

Contribution

It presents a novel algorithm for deriving the subgroup pattern of cyclic extensions from known patterns, enabling computation for broader classes of groups.

Findings

Algorithm successfully computes subgroup patterns of cyclic extensions.

Method extends to solvable groups via composition series.

Facilitates analysis of subgroup structures in complex groups.

Abstract

The subgroup pattern of a finite groups $G$ is the table of marks of $G$ together with a list of representatives of the conjugacy classes of subgroups of $G$. In this article we present an algorithm for the computation of the subgroup pattern of a cyclic extension of $G$ from the subgroup pattern of $G$. Repeated application of this algorithm yields an algorithm for the computation of the table of marks of a solvable group $G$, along a composition series of $G$.