Automorphism group of an abelian group

Solution 1:

If G is a finite direct product of cyclic groups of order pⁿ, then Aut(G) is isomorphic to the group GL(n,Z/pⁿZ) of invertible n×n matrices over the ring Z/pⁿZ. These matrices work surprisingly similarly to matrices over a field. They are invertible if and only if their determinant is invertible (so not divisible by p).

G is called a homocyclic group. Aut(G) has a normal subgroup consisting of those matrices which act trivially on the vector space G/G^p. This normal subgroup is called a congruence subgroup and consists of those matrices that are congruent to the identity mod p.

General finite abelian groups have slightly more complicated automorphism groups, which are still matrix groups, but where the entries come from different rings and bi-modules (that is, there are different sorts of congruence conditions required). Similar techniques work for understanding finite-dimensional algebras.

Solution 2:

You're right - it's not true (why would it be, really?). To see that it's enough to consider a single counterexample - just take $p=n=2$. The group $\mbox{Aut}(\mathbb{Z}_4 \times \mathbb{Z}_4)$ is of order 96, while $\mbox{GL}(2,4)$ is of order 180.