The multiple holomorph of centerless groups

Abstract

Let G be a group. The holomorph $\mathrm{Hol}\left(G\right)$ of G may be defined as the normalizer of the subgroup of either left or right translations in the group of all permutations of G. The multiple holomorph $\mathrm{NHol}\left(G\right)$ is in turn defined as the normalizer of the holomorph. Their quotient $T\left(G\right)=\mathrm{NHol}\left(G\right)/\mathrm{Hol}\left(G\right)$ has been computed for various families of groups G. In this paper, we consider the case when G is centerless, and we show that $T\left(G\right)$ must have exponent at most 2 unless G satisfies some fairly strong conditions. As applications of our main theorem, we are able to show that $T\left(G\right)$ has order 2 for all almost simple groups G, and that $T\left(G\right)$ has exponent at most 2 for all centerless perfect or complete groups G.