Zeta functions and subgroup growth in $P2/m$

TL;DR

This paper uses zeta functions of space groups to precisely count subgroups and normal subgroups of a specific crystallographic group, leading to insights on its subgroup growth and structure.

Contribution

It introduces a method to compute subgroup counts via zeta functions for the tenth crystallographic group, advancing understanding of subgroup growth in space groups.

Findings

Exact subgroup counts for the tenth crystallographic group

Analysis of subgroup growth degree

Insights into the structure of the group

Abstract

By means of zeta and normal zeta functions of space groups, we determine the number of subgroups, resp. normal subgroups, of the tenth crystallographic group for any given index. This enables us to draw conclusions on the subgroup growth and the degree of this group.