Automorphism Group Of Direct Product Of Finite Groups

Let G = H×K be the direct product of finite groups H and K. Determiningthe structure of the automorphism group Aut G of G from H and K is a fundamental and important problem. In the paper published by Bidwell, Curran and McCaughan in 2006, four special subgroups A, B, C, D of Aut G were defined firstly, which satisfyand then an important result (called the BCM Theorem in this paper) that Aut G = ABCD if H and K have no common direct factor was proven.In this paper we first obtain a necessary and sufficient condition for Aut G = ABCD.Theorem 1. Let G = H×K be the direct product of two finite groups. ThenAut G = ABCD if and only if H^?∩K = 1 for all (?)∈Aut G.Theorem 1 not only implies the BCM Theorem immediately (see Corollary 1 in this paper), but also deduces the following useful corollary:Corollary 2. Let G = H×K be the direct product of two finite groups. If H or K is a characteristic subgroup of G, then Aut G = ABCD.In general, for the direct product G = G₁×…×G_n and a map (?): G→G, the mapmatrix M((?)) = ((?)_ij)_n×n is defined in the paper to characterize all the endomorphismsand automorphisms of G.Theorem 2. Let G = G₁×…×G_n be the direct product of finite groups. Thena map (?) : G→G is an endomorphism of G if and only if the associated map matrixM((?)) satisfies the following conditions:(1)(?)_ij∈Hom(G_i,G_j),(?)1≤i,j≤n,(2)[Im(?)_ij,Im(?)_kj]=1,(?)1≤i,j,k≤n but i≠k.Theorem 3. Let G = G₁×…×G_n be the direct product of finite groups. If G_iand G_j have no common direct factor for any 1≤i < j≤n, then a map (?) : G→Gis an automorphism of G if and only if the associated map matrix M((?)) satisfies the following conditions:(1)(?)_ii∈Aut G_i,1≤i≤n,(2)(?)_ij∈Hom(G_i,Z(G_j)),1≤i≠j≤n.Conversely, the matrix representation of each automorphism of G satisfies the above two conditions. As an application of Theorem 3, we obtain an order formula of the automorphism group of the direct product of groups.Corollary 3. Let G =G₁×…×G_n be the direct product of finite groups. If G_i and G_j have no common direct factor for any 1≤i < j≤n, thenFinally, the order of the automorphism group of a finite abelian p-group is given immediately by Corollary 3.