On pronormality of subgroups of odd index in finite groups

Abstract: In this talk we discuss a recent progress in research of pronormality of subgroups of odd index in finite groups.

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 $\langle H, H^g \rangle$ 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, remarkable results by L. Babai, P. Palfy, Ch. Praeger, and others). Thus, the question of description of families 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.

In 2012, E.P. Vdovin and D.O. 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.S. Kondrat'ev, the speaker, and D. Revin in 2016. However, in many finite simple groups the subgroups of odd index are pronormal. Moreover, the question of pronormality of a subgroup of odd index in an arbitrary finite group can be partially reduced to questions of pronormality of some subgroups of odd indices in its chief factors.

This talk is partially based on joint results with S. Glasby, A.S. Kondrat’ev, C.E. Praeger, and D.O. Revin.

group theory

Audience: researchers in the topic

GOThIC - Ischia Online Group Theory Conference

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