Isomorphism in expanding families of indistinguishable groups

TL;DR

The paper constructs large families of indistinguishable yet nonisomorphic groups with identical character tables and automorphism properties, and provides a polynomial-time isomorphism testing algorithm.

Contribution

It introduces a new class of groups that are virtually indistinguishable but nonisomorphic, along with an efficient isomorphism testing method.

Findings

Families of groups with identical character tables and automorphism structures.

Existence of a polynomial-time algorithm for group isomorphism testing.

Construction of large nonisomorphic groups with controlled properties.

Abstract

For every odd prime $p$ and every integer $n\geq 12$ there is a Heisenberg group of order $p^{5n/4+O(1)}$ that has $p^{n^2/24+O(n)}$ pairwise nonisomorphic quotients of order $p^{n}$. Yet, these quotients are virtually indistinguishable. They have isomorphic character tables, every conjugacy class of a non-central element has the same size, and every element has order at most $p$. They are also directly and centrally indecomposable and of the same indecomposability type. The recognized portions of their automorphism groups are isomorphic, represented isomorphically on their abelianizations, and of small index in their full automorphism groups. Nevertheless, there is a polynomial-time algorithm to test for isomorphisms between these groups.