Automorphism groups of direct products of non-finite cyclic groups

Question

I'd like to learn more about the automorphism group of direct products of cyclic groups where at least one factor is $\mathbb Z$. There seem to be many articles studying analogous questions for finite groups, but i am unable to find anything for products infinite groups, not even for cyclic ones. Does anyone know any references that study this or something more general?

Answer 1

The analysis is the same, whether the groups are finite or infinite.

Suppose $G$ and $H$ are two abelian groups. A morphism $\varphi\colon G\times H\to G\times H$ corresponds to four morphisms, $$\begin{align*} \varphi_{G,G}\colon &\ G\to G\\ \varphi_{G,H}\colon &\ G\to H\\ \varphi_{H,G}\colon &\ H\to G\\ \varphi_{H,H}\colon &\ H\to H \end{align*}$$ by the rule that $$\varphi(g,h) = \Bigl( \varphi_{G,G}(h)\varphi_{H,G}(h), \varphi_{G,H}(g)\varphi_{H,H}(h)\Bigr);$$ namely, $\varphi_{G,G}$ is the restriction of $\pi_G\circ\varphi$ to $G$; $\varphi_{G,H}$ is the restriction of $\pi_G\circ\varphi$ to $G$, etc.

We can view this as a $2\times 2$ matrix acting on a column vector, $$\left(\begin{array}{cc} \varphi_{G,G} & \varphi_{H,G}\\ \varphi_{G,H} & \varphi_{G,G}\end{array} \right)\left(\begin{array}{c}g\\h\end{array}\right) = \varphi\left(\begin{array}{c}g\\h\end{array}\right).$$

It is now straightforward to verify that if $\varphi$ and $\theta$ are morphisms, then the matrix corresponding to $\varphi\circ\theta$ is precisely the "product" of the matrices corresponding to $\varphi$ and to $\theta$. So that the endomorphisms of $G\times H$ correspond to $$\left(\begin{array}{cc} \mathrm{Hom}(G,G) & \mathrm{Hom}(H,G)\\ \mathrm{Hom}(G,H) & \mathrm{Hom}(H,H) \end{array}\right).$$ When one of the factors is torsion (say $H$) and the other is torsionfree (say $G$), then one of the off-diagonal entries is necessarily trivial (in this case, $\mathrm{Hom}(H,G)$). And then it is straightforward to check that the endomorphism is an automorphism if and only if the diagonal entries are automorphisms.

For example, $\mathrm{Aut}(\mathbb{Z}\times C_2)$ will then correspond to matrices of the form $$\left(\begin{array}{cc} \psi & 0\\ \theta & 1\end{array}\right),$$ where $\psi$ is an automorphism of $\mathbb{Z}$ (either the identity, or the map that sends $a$ to $-a$), $1$ is the identity map of $C_2$ (its only automorphism), and $\theta\colon\mathbb{Z}\to C_2$ is a morphism; either the map that sends everything to $0$, or else the map that sends $1\in\mathbb{Z}$ to the generator of $C_2$. So you have a total of four automorphisms.

The automorphisms of $\mathbb{Z}\times C_4$, on the other hand, correspond to matrices of the form $$\left(\begin{array}{cc} \psi & 0\\ \theta & \phi \end{array}\right)$$ where $\psi$ is an automorphism of $\mathbb{Z}$ (either the identity or multiply-by-minus-one); $\phi$ is an automorphism of $C_4$ (either the identity, or the map sending each element to its inverse), and $\theta\colon\mathbb{Z}\to C_4$ is any of the four possible morphisms, sending $1$ to your favorite element of $C_4$. So here we have $2\times 2\times 4 = 16$ possible automorphisms.

For three or more factors, provided there are only finitely many factors, the answer is similar: the endomorphisms of $A_1\times\cdots\times A_n$ correspond to $n^2$ morphisms, $\varphi_{i,j}\colon A_i\to A_j$, and you can "arrange" them in a matrix form so that composition of endomorphisms corresponds to multiplication of matrices, and the automorphisms are precisely the invertible matrices. Having some groups be torsion and some torsionfree means that some of the entries must be equal to $0$ (the ones corresponding to morphisms from the torsion groups to the torsionfree groups).