SEMINAR: Groups and Combinatorics Seminar: Natal'ya Maslova, 4pm Oct 25 in Weatherburn LT

Speaker: Natalia Maslova (Russian Academy of Sciences, Krasovsky Institute of Mathematics and Mechanics and Ural Federal University)

Title: On pronormality of subgroups of odd index in finite groups

Time and place: 4pm Friday 25 Oct 2019, Weatherburn LT

Abstract: A subgroup H of a group G is pronormal in G if for any element g from G, subgroups H and H^g are conjugate in the subgroup generated by H and H^g. Some problems in finite group theory, combinatorics, and permutation group theory were solved in terms of pronormality (see, for example, results by L. Babai, P. Palfy, Ch. Praeger, and others). Thus, the question of description of pronormal subgroups in finite groups is of interest. Well-known examples of pronormal subgroups in finite groups are normal subgroups, maximal subgroups, Sylow subgroups, Carter subgroups, Hall subgroups of solvable groups, and so on. An important problem is to describe families of pronormal subgroups in finite simple groups.

In 2012, E. Vdovin and D. Revin proved that the Hall subgroups are pronormal in finite simple groups and conjectured that the subgroups of odd index are pronormal in finite simple groups. This conjecture was disproved by A. Kondrat'ev, the speaker, and D. Revin in 2016. However, in many finite simple groups the subgroups of odd index are pronormal. In this talk we discuss a recent progress in the classification of finite simple groups in which the subgroups of odd index are pronormal and some connected questions.