$2$-local amalgams for the simple groups ${\rm GL}(5,2), {\rm M}_{24}$ and ${\rm He}$

Abstract

We elaborate on a method of G. Michler [Mi] to construct finite groups with a prescribed involution centralizer $H$ using compatible pairs and amalgamation of $H$ and another naturally arising $2$-local subgroup. Here we deal with the particular set-up leading to the simple groups $GL(5,2), M_{24}$ and $He$.