Automorphism group of a finite group

Question

I would like to ask if there exists an explicit description of $\mathrm{Aut}(G)$, the group of automorphisms of a finite group $G$, in particular, when $G$ is abelian. E.g., if $G = \mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/m\mathbb{Z}$, where $m$ is a positive integer, how can we describe $\mathrm{Aut}(G)$? Which relation do we have between it and $GL_m(\mathbb{Z})$? If $m$ is prime then $\mathrm{Aut}(G) \cong GL_m(\mathbb{Z})$, but what happens for $m$ general? E.g., if $m=4$, I find that the cardinality of $\mathrm{Aut}(\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z})$ is $8$, by seeing to where the generators $(0,1)$ and $(1,0)$ are sent. But I have the feeling that $GL_4(\mathbb{Z})$ should be contained in $\mathrm{Aut}(\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z})$, so I should have a problem with the cardinalities.

More general, is it sufficient to see where a minimal set of generators is sent? Maybe someone could indicate me a paper where I could find a good description of such automorphism groups?

Answer 1

A nice set of generators for the automorphism group of a finite abelian group is described by Garrett Birkhoff in his paper titled "Subgroups of abelian groups", Proc. London Math. Soc., s2-38(1):385-401, 1935 (MR).

Note that each finite abelian group is the product of its $p$-primary subgroups. An automorphism preserves the primary parts. So it is sufficient to consider $p$-abelian groups.

It is convenient to think of automorphisms of finite abelian groups as integer matrices. Expressing the group $A = \mathbf Z/p^{\lambda_1}\oplus \dotsb \oplus \mathbf Z/p^{\lambda_n}$ as a quotient of the free abelian group $\mathbf Z^n$, lift an automorphism $\phi$ of $A$ to an automorphism $\tilde\phi$ of $\mathbf Z^n$: $$ \begin{matrix} \mathbf Z^n & \xrightarrow{\tilde \phi} & \mathbf Z^n\\ \downarrow & &\downarrow\\ A & \xrightarrow{\phi} & A \end{matrix} $$ The matrix $(\phi_{ij})$ representing $\tilde\phi$ is an invertible integer matrix. As far as the automorphism $\phi$ is concerned, the its entries in the $i$th row are in $\mathbf Z/p^{\lambda_i}\mathbf Z$. Also $\phi_{ij}$ is divisible by $p^{\max(0, \lambda_j-\lambda_i)}$. With this matrix representation, it is easy to do calculations. For example, composition is matrix multiplication.

As for your query concerning cardinalities: if $\lambda_1>\lambda_2>\dotsb>\lambda_n$ and $$ A = (\mathbf Z/p^{\lambda_1}\mathbf Z)^{\oplus m_1}\oplus\dotsc \oplus (\mathbf Z/p^{\lambda_n}\mathbf Z)^{\oplus m_n}, $$ then it is possible to deduce from the above description of the automorphism group that $$ \lvert\mathrm{Aut}(A)\rvert = q^{\sum_{i,j}m_im_j\min(\lambda_i,\lambda_j)}\prod_{k=1}^n \prod_{l=1}^{m_k} (1 - q^{-l}). $$ The group $\prod_{k=1}^n GL_{m_k}(\mathbf Z/p\mathbf Z)$ is a quotient of $\mathrm{Aut}(A)$ by a $p$-group.