I'm reading the proof of a theorem that says
If $G$ is a finitely generated FC-group, then its set of torsion elements $T(G)$ is a finite normal subgroup of $G$.
I understand everything the proof says, but I don't understand why it doesn't explain why $T(G)$ is finite. Is it obvious that a finitely generated group has only finitely many torsion elements?
EDIT: Thanks to Jim Belk's answer, I know that being finitely generated isn't enough to have finitely many torsion elements for groups. Why is that true for FC-groups then? I know and can prove that that the commutator subgroup of a finitely generated FC-group is finite, but in such groups $G'\subseteq T(G)\subseteq G$, so this doesn't immidiately imply the finiteness of $T(G)$…
Best Answer
A finitely generated group can have infinitely many torsion elements. For example, the infinite dihedral group $\langle a,b \mid a^2=b^2=1\rangle$ has infinitely many elements of order two.