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Algebra Seminar
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Alexander Ivanov
Imperial College

Tri-extraspecial groups

Friday 26th March, 12:05-12:55pm, Carslaw 373.

The talk is based on a joint work with Sergey V. Shpectorov on a class of groups, which are certain split extensions of special 2-groups 23+6n by L3(2). We call these groups tri-extraspecial, because they behave very much like extraspecial 2-groups. For every value of n, there are exactly two such groups, denoted T±(n). We show that the outer automorphism group of T±(n) is 2 x Sp(2n,2). The group Out(T±(n)) acts transitively on the classes of complements L3(2) in T±(n), the stabilizer of such a class being O±(2n,2). The group T±(n) arises as a normal subgroup in a maximal parabolic subgroup (the stabilizer of a 3-dimensional totally singular subspace) of O±(2n+6,2). Even more remarkably, the groups T+(4) and T-(4) arise, respectively, in the sporadic groups J4 and F24. These latter examples were the primary motivation of our interest in tri-extraspecial groups.