Automorphisms of a Finite Abelian Group Which Reduce to the Identity on a Subgroup or Factor Group

1. The automorphism group of any group G we denote by Y(G). We shall consider the following two types of subgroups of T(G) : if H is a subgroup of G, Y(G: H) is the subgroup consisting of all automorphisms which are the identity on H, and if H is a normal subgroup, Y(G: G/H) is the subgroup consisting of all automorphisms which leave H invariant and which induce the identity on G/H. For the special case where G is finite Abelian, it is the connection between these two types of groups which is to be studied in this paper. Since any finite Abelian group is the direct sum of ¿»-groups, each of which is a characteristic subgroup, it will then be sufficient to consider only the case of a /»-group, for some fixed prime p. We will write the group additively. The order of any element g of a finite group G we denote by \g\. All operators are to be written on the right. It may perhaps be mentioned here that these subgroups of automorphisms of an Abelian group do arise in connection with the theory of the structure of the automorphism group of any finite group G. In fact, it has been shown by Green [3], that if G has centre Z and derived group G' then, writing Tl = G/Z, there exists an Abelian group Q containing an isomorphic copy of Z with Q/Z=A1/I1', such that