The multiple holomorph of centerless groups

Abstract: The holomorph $\operatorname{Hol}(G)$ of a group $G$ may be defined as the normalizer of the subgroup of left translations in the group of all permutations of $G$. The multiple holomorph $\operatorname{NHol}(G)$ of $G$ may in turn be defined as the normalizer of the holomorph. Their quotient $T(G) = \operatorname{NHol}(G)/\operatorname{Hol}(G)$ has been computed for various families of groups G, and interestingly $T(G)$ turns out to be elementary $2$-abelian in many of the known cases. In this talk, we consider the case when $G$ is centerless, and we will present our new result that $T(G)$ has to be elementary $2$-abelian unless G satisfies some fairly strong conditions. For example, our result implies that T(G) is elementary $2$-abelian when $G$ is any (not necessarily finite) centerless perfect/almost simple/complete group.

group theory

Audience: researchers in the topic

GOThIC - Ischia Online Group Theory Conference

Series comments: Please send a message to andrea.caranti@unitn.it to receive a link to the Zoom room where the conference (a series of talks, actually) takes place.