Groups of order $p^9$, class 2, and exponent $p$ having derived group of order $p^2$

TL;DR

This paper classifies all special groups of order p^9 with derived group of order p^2, introduces a digraph representation for these groups, and provides invariants to distinguish their isomorphism types.

Contribution

It identifies six isomorphism types of such groups and develops a novel digraph-based method for representing and analyzing class 2 groups of exponent p.

Findings

Six isomorphism types of groups of order p^9 with derived group of order p^2 identified.

A digraph representation encodes structural information of these groups.

Invariants derived from digraphs help distinguish non-isomorphic groups.

Abstract

This paper concerns finite groups of class (at most) two and of odd prime exponent $p$. Such a group is called special if the center lies within its derived group. Every group of class 2 and exponent $p$ can be uniquely expressed as the direct product of an elementary abelian group and a special group. This reduces the isomorphism problem to special groups. The special groups having $G'$ cyclic are well known. Groups having $|G'|=p^2$ are known to be generated by two abelian subgroups. As such, they can be described by a pair of Scharlau Matrices which we will define. Using these, Vishnevetskii ([1], [2]) classified the special groups which are not central products of groups of smaller order. We call these Vishnevetskii indecomposable. All decomposable groups are central products of two or more indecomposable groups. By Theorem 2 of [2], if $G$ is a special group with derived group of order $p^2$ then the indecomposable central factors of $G$, together with their multiplicities, form a set of invariants for $G$. However these invariants do not determine $G$, since (as we will show below) two nonisomorphic groups can have the same indecomposable central factors. The groups of exponent $p$ and order dividing $p^8$ are already in the literature. In this paper we prove that there are six isomorphism types of special groups of order $p^9$ having $|G'|=p^2$, and we list them. We will also introduce a way of representing groups of class 2 having exponent $p$ by using digraphs with flows on the edges. The digraph is compact and gives, on sight, a great deal of structural information about the group and completely defines the group. Some group invariants will also be described which are easy to compute from the digraphs and which can readily be used to distinguish the isomorphism types for small orders.