Suppose that \(G\) is a finitely generated group. Let \(n\) be a positive integer. Prove that \(G\) has only finitely many subgroups of index \(n\)
Let \(p\) be a prime number. If \(G\) is any finitely-generated abelian group, let \(t_p(G)\) denote the number of subgroups of \(G\) of index \(p\). Determine the possible values of \(t_p(G)\) as \(G\) varies over all finitely-generated abelian groups.
Solution
Goal. (1) A finitely generated group has finitely many subgroups of index \(n\). (2) Determine the possible values of \(t_p(G)\) for finitely generated abelian \(G\).
1. (1) \(G\) has finitely many subgroups of index \(n\).
1.1. A subgroup \(H \le G\) of index \(n\) corresponds to a transitive action of \(G\) on a set of \(n\) elements (the cosets \(G/H\)).
\(G\) acts on \(G/H\) by left multiplication, transitively, with stabilizer \(H\).
1.2. A transitive action of \(G\) on an \(n\)-element set is determined by a homomorphism \(G \to S_n\) (up to conjugacy).
the action is a homomorphism \(G \to S_n\), and the stabilizer of a point determines the action up to conjugacy.
1.3. There are finitely many homomorphisms \(G \to S_n\).
\(G\) is finitely generated, say by \(g_1, \dots, g_m\); a homomorphism \(G \to S_n\) is determined by the images of the generators, and there are \(|S_n|^m = (n!)^m\) choices.
1.4. Hence there are finitely many subgroups of index \(n\).
each subgroup of index \(n\) gives a homomorphism \(G \to S_n\) (the action on cosets), and finitely many homomorphisms give finitely many subgroups.
2. (2) The possible values of \(t_p(G)\).
2.1. Subgroups of index \(p\) in an abelian group correspond to surjections \(G \to \ZZ/p\).
a subgroup \(H\) of index \(p\) gives \(G/H \cong \ZZ/p\) (the only group of order \(p\)), and conversely the kernel of a surjection \(G \to \ZZ/p\) has index \(p\).
2.2. The number of surjections \(G \to \ZZ/p\) is \(p^r - 1\) where \(r = \dim_{\ZZ/p}(G/pG)\).
\(\operatorname{Hom}(G, \ZZ/p) \cong \operatorname{Hom}(G/pG, \ZZ/p) \cong (\ZZ/p)^r\), and the nonzero homomorphisms (surjections) number \(p^r - 1\).
2.3. \(r = \operatorname{rank}(G) + (\text{number of } p\text{-torsion cyclic factors})\).
\(G \cong \ZZ^s \oplus (\text{torsion})\), and \(G/pG \cong (\ZZ/p)^s \oplus (\text{torsion}/p\text{-torsion})\), so \(r = s + t\) where \(t\) is the number of cyclic \(p\)-power torsion summands.
2.4. Hence \(t_p(G) = p^r - 1\) for some \(r \ge 0\), and every value of the form \(p^r - 1\) (\(r \ge 0\)) occurs.
\(t_p(G) = p^r - 1\) by 2.2, and \(r\) can be any nonnegative integer (take \(G = \ZZ^r\)).
3. Q.E.D.
1 proves (1); 2 shows \(t_p(G) \in \theset{p^r - 1 : r \ge 0}\).