Speaker: James B. Wilson, Colorado State University
Abstract: Attacks on the Group Isomorphism Problem have diversified in the last decade with approaches ranging from Erd˝os-R´enyi type models, to hyper-graph isomorphism techniques, and several representation and Lie theory methods. We will summarize necessary foundations and highlight important recent progress while concentrating on examples.
As our main illustration, we will use a method from the study of groups of finite Morley rank. We construct a family of non-isomorphic groups that share all the same proper subgroups, all the same proper quotient groups, the same character tables, and the same automorphism groups. This settles a question of Gowers and Babai about the minimum requirements to decide isomorphism. And yet, we also prove a polylogarithmic isomorphism test for this indistinguishable family of groups. As these and other examples demonstrate we have a lot yet to learn about most groups.
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