Computing canonical labellings of finite solvable groups

Abstract

We define a canonical labelling function on the class of finite solvable groups so that two such groups $G$ and $H$ are isomorphic if and only if can$(G)=$can$(H)$. Specifically, can$(G)$ is a group presentation that describes a group isomorphic to $G$, and our description explains how to construct an isomorphism $G\to$can$(G)$. Our approach is motivated by O'Brien's (1993) canonical presentations for finite $p$-groups and utilises ideas from group cohomology first described by Robinson (1982) and automorphism group algorithms developed by Smith (1994), Holt (2001), and others. We also discuss a proof-of-concept implementation for the computer algebra system GAP and comment on the major bottlenecks and open research questions.