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Automorphisms of nilpotent groups and nonassociative rings

SPEAKER: James Wilson

TITLE: Automorphisms of nilpotent groups and nonassociative rings (joint work with U. First of The Hebrew University).

ABSTRACT: Since the 1930’s in work by Hall and Baer we have understood that automorphisms of p-groups are controlled largely by the action of linear groups on tensor products. But this has offered mostly vocabulary and little practical understanding. So we consider what category describes the problem and to our surprise we find not one, but 52 natural categories control the automorphisms of a p-group (and also associative and Lie algebras). We organize this into a single 2-category. The abstraction exposes 4 rings that heavily constrain the action of automorphism groups of p-groups. For certain classes of rings, e.g. Azumaya rings, the automorphism groups can be characterized. Though abstractly inspired the outcome is highly concrete and leads to polynomial time algorithms in many critical cases including “random” p-groups. Generalizations to multilinear products also follow.