Siblings and twins in finite p-groups and a group identification for the groups of order $2^9$

TL;DR

This paper explores invariants for distinguishing p-groups, introduces concepts of siblings and twins to identify difficult cases, and develops an algorithm to uniquely identify over ten million groups of order 2^9.

Contribution

It introduces the notions of siblings and twins for p-groups and creates an effective algorithm for identifying all groups of order 2^9.

Findings

Identification of siblings and twins among small p-groups

Development of an algorithm for group identification of order 2^9

Successful distinction of over 10 million groups of order 2^9

Abstract

We investigate which invariants of groups are powerful in distinguishing non-isomorphic p-groups. We introduce the notations of siblings and twins for p-groups that are difficult to distinguish and we describe the siblings and twins among the groups of small prime-power order. We then use these ideas to device an effective group identification algorithm for the $10,494,213$ groups of order $2^9$.