A characterisation of A-simple groups

Let $A$ be an elementary abelian $r$-group with rank at least $3$ that acts faithfully on the finite $r^{\mathrm{\prime }}$-group $G$. Assume that $G$ is $A$-simple, so that $G=K_1\times \cdots \times K_n$ where $K_1,\dots ,K_n$ is a collection of simple subgroups of $G$ that is permuted transitively by $A$. The purpose of this paper is to characterize $G$ and the collection of fixed point subgroups $\{C_G(a)\mathrm{\mid }a\in A^{\mathrm{\#}}\}$. An application of this result will be a new proof of McBride’s Nonsolvable Signalizer Functor Theorem.