Characteristic Subgroups of Finite Abelian Groups

We consider the question: When do two finite abelian groups have isomorphic lattices of characteristic subgroups? An explicit description of the characteristic subgroups of such groups enables us to give a complete answer to this question in the case where at least one of the groups has odd order. An “exceptional” isomorphism, which occurs between the lattice of characteristic subgroups of Zp × Zp2 × Zp4 and Zp2 × Zp5, for any prime p, is noteworthy. In 1939, Baer [2] considered the question: When do two groups have isomorphic lattices of subgroups? Since in general this is a very difficult problem, Baer restricted his attention primarily to the case of abelian groups. Even in this case, a complete solution has only very recently been obtained, in [6]. Most of the complications arise in the case where both groups are infinite of torsion-free rank 1. In particular, if both groups are finite, the situation is fairly uncomplicated; the following theorem, which provides a complete solution to the problem in this case, follows immediately from Theorem 1.1(b) of [6], where the result is credited to Baer: Theorem. Let G and H be two finite abelian groups. Then G and H have isomorphic lattices of subgroups if and only if there is a bijection φ from the set of Sylow subgroups of G to the set of Sylow subgroups of H such that for all Sylow subgroups P of G, (i) If P is cyclic of order p for prime p, then φ(P ) is cyclic of order q for some prime q. (ii) If P is not cyclic, then φ(P ) ∼= P . Given this success, it seems natural to consider a related question: When do two groups have isomorphic lattices of characteristic subgroups? Again, the general problem seems to be very difficult. We will consider only the case of finite abelian groups. We show in §4 that this problem can be reduced to the case in which both groups are abelian p-groups (for the same prime p). Our main result then gives a solution in the case p 6= 2: