Determine all finitely generated abelian groups \(G\) for which \[ \operatorname{Aut}(G) \] is finite.
Solution
By the structure theorem for finitely generated abelian groups, write \[ G\cong\ZZ^r\oplus T, \] where \(r\geq0\) and \(T\) is finite.
1. If \(r\geq2\), then \(\operatorname{Aut}(G)\) is infinite.
Let \(e_1,e_2,\ldots,e_r\) be the standard basis of the free summand. For each \(n\in\ZZ\), define an automorphism of \(\ZZ^r\oplus T\) by \[ e_1\longmapsto e_1+n e_2, \qquad e_j\longmapsto e_j\quad(j\geq2), \] and let it act as the identity on \(T\). Its inverse is obtained by replacing \(n\) with \(-n\). Distinct integers \(n\) give distinct automorphisms, so the automorphism group is infinite.
2. If \(r=0\), then \(\operatorname{Aut}(G)\) is finite.
In this case \(G=T\) is a finite set. Hence the set of all maps \(G\to G\) is finite, and \(\operatorname{Aut}(G)\) is a subset of it.
3. Suppose \(r=1\), so\[ G\cong\ZZ e\oplus T. \]Every automorphism \(\varphi\) has the form\[ \varphi(e)=\epsilon e+t, \qquad \varphi|_T=\alpha, \]where\[ \epsilon\in\{1,-1\}, \qquad t\in T, \qquad \alpha\in\operatorname{Aut}(T). \]
The torsion subgroup \(T\) consists intrinsically of all elements of finite order, so it is characteristic. Thus any automorphism \(\varphi\) restricts to an automorphism \[ \alpha=\varphi|_T \] of \(T\) and induces an automorphism of \[ G/T\cong\ZZ. \] Every automorphism of \(\ZZ\) is multiplication by \(1\) or \(-1\). Therefore, for some \(\epsilon\in\{1,-1\}\), \[ \varphi(e)+T = \epsilon e+T, \] which means \[ \varphi(e)=\epsilon e+t \] for some \(t\in T\).
4. If \(r=1\), then \(\operatorname{Aut}(G)\) is finite.
By step 3, an automorphism is determined by the three pieces of data \[ \epsilon\in\{1,-1\}, \qquad t\in T, \qquad \alpha\in\operatorname{Aut}(T). \] All three sets of choices are finite because \(T\) is finite. Thus \(\operatorname{Aut}(G)\) is finite.
5. Consequently, the required groups are exactly the finitelygenerated abelian groups of free rank at most one:\[ \boxed{ G\cong T \quad\text{or}\quad G\cong\ZZ\oplus T, \qquad T\text{ finite}. } \]
Steps 2 and 4 prove finiteness when \(r\leq1\), while step 1 proves infinitude when \(r\geq2\).
6. Q.E.D.
Step 5 gives the complete classification.