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Semiprimitive groups - a classification theorem (sort of)

Banff International Research Station for Mathematical Innovation and Discovery

A transitive permutation group is called semiprimitive if each normal subgroup is transitive or semiregular. This large class of groups includes the classes of primitive, quasiprimitive, innately transitive and Frobenius groups. Apart from being a generalisation of these important classes of permutation groups, motivation to study this class came from problems in abstract algebra and in algebraic graph theory. A barrier to their study has been the lack of any apparent structure and the prevalence of wild examples. In this talk I will report on joint work with Michael Giudici in which we brought some clarity to the study of this class of groups. We found that there is strong structure to a semiprimitive group, although not as precise as the O'Nan-Scott Theorem for primitive groups and there is rough structure that explains how semiprimitive groups are built from innately transitive groups. Along the way I'll mention plenty of examples and time permitting some application of this theory to the motivating problems.

Mathematics; Group theory and generalizations; Topological groups, lie groups; Algebraic structures

video/mp4

Author affiliation: The University of Western Australia

2017-05-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61678

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/