Weisfeiler-Leman for Group Isomorphism: Action Compatibility
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In this paper, we show that the constant-dimensional Weisfeiler-Leman algorithm for groups (Brachter Schweitzer, LICS 2020) serves as a unifying isomorphism test for several families of groups that were previously shown to be in 𝖯 by multiple methods. This includes coprime extensions where the normal Hall subgroup is Abelian and the complement is O(1)-generated (Qiao, Sarma, Tang, STACS 2011), and groups with no Abelian normal subgroups (Babai, Codenotti, Qiao, ICALP 2012). In both of these cases, the previous strategy involved identifying key group-theoretic structure that could then be leveraged algorithmically, resulting in different algorithms for each family. A common theme among these is that the group-theoretic structure is mostly about the action of one group on another. Our main contribution is to show that a single, combinatorial algorithm (Weisfeiler-Leman) can identify those same group-theoretic structures in polynomial time. We also derive consequences for the parallel complexity of testing isomorphism in these families. In particular, we show that constant-dimensional Weisfeiler-Leman only requires a constant number of rounds to identify groups in the above classes. Combining this result with the parallel Weisfeiler-Leman implementation of Grohe Verbitsky for graphs (ICALP 2006), this puts isomorphism testing for each of these families of groups into 𝖳𝖢^0. However, we show that count-free Weisfeiler-Leman is not strong enough to put isomorphism testing even for Abelian groups into 𝖠𝖢^0.
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