2025/04/28 by Joshua A. Grochow, D. Johnson, Grochow, Joshua A. +3 · 1 citation
Engineering · Computer Science · Mathematics · #graph theory and CDMA systems #Advanced Graph Theory Research #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2504.19777
In this paper, we exhibit an \textsfAC3 isomorphism test for groups without Abelian normal subgroups (a.k.a. Fitting-free groups), a class for which isomorphism testing was previously known to be in P (Babai, Codenotti, and Qiao; ICALP '12). Here, we leverage the fact that G/PKer(G) can be viewed as permutation group of degree O(log |G|). As G is given by its multiplication table, we are able to implement the solution for the corresponding instance of Twisted Code Equivalence in \textsfAC3. In sharp contrast, we show that when our groups are specified by a generating set of permutations, isomorphism testing of Fitting-free groups is at least as hard as Graph Isomorphism and Linear Code Equivalence (the latter being \textsfGI-hard and having no known subexponential-time algorithm). Lastly, we show that any Fitting-free group of order n is identified by \textsfFO formulas (without counting) using only O(log log n) variables. This is in contrast to the fact that there are infinite families of Abelian groups that are not identified by \textsfFO formulas with o(log n) variables (Grochow & Levet, FCT '23).