Reduction of the group isomorphism problem to the group automorphism problem
Saveliy V. Skresanov
TL;DR
This work establishes that the group isomorphism problem can be solved in polynomial time by reducing it to the group automorphism problem, mirroring the known graph-theoretic reductions. It presents three main reductions to GrpACOUNT, GrpAPART, and GrpAGEN, leveraging a detailed structural description of Aut(G×H) when the input groups are nonisomorphic (and a swap automorphism when they are). Core tools include a poly-time direct-factor decomposition, Krull–Schmidt uniqueness, and the automorphism-group description of a direct product, enabling poly-time counting of Hom spaces and related automorphism counts. The paper proves several polynomial-time equivalences among GrpISO, GrpACOUNT, GrpAGEN, GrpAPART, GrpIMAP, and GrpICOUNT (with colored-group variants discussed), thereby extending the graph-isomorphism paradigm to group theory and informing automorphism-based approaches to group isomorphism testing.
Abstract
It is well known that the graph isomorphism problem is polynomial-time reducible to the graph automorphism problem (in fact these two problems are polynomial-time equivalent). We show that, analogously, the group isomorphism problem is polynomial-time reducible to the group automorphism problem. Reductions to other relevant problems like automorphism counting are also given.