TITLE: Subgroups of free products of cyclic groups
QUESTION [6 upvotes]: Consider the free product $\mathbb{Z}_{3} \star \mathbb{Z}_{3}$. How would one determine the number of subgroups of this product up to isomorphism? It is routine for the case of the product $\mathbb{Z}_{2} \star \mathbb{Z}_{2}$ either by considering the combinations of elements from each group separately or the number of covering spaces of the wedge product of projective spaces. I feel that the easiest way would be to consider the subgroups from the group directly without resorting to topology, but how would one do this?

REPLY [4 votes]: There is a generalization, and it is called "Bass-Serre theory". What you learn from that theory is that $\mathbb{Z}_3 * \mathbb{Z}_3$ is a considerably more complicated group than $\mathbb{Z}_2 * \mathbb{Z}_2$. Every infinite subgroup of $\mathbb{Z}_2 * \mathbb{Z}_2$ is isomorphic to either itself or to the infinite cyclic group, whereas $\mathbb{Z}_3 * \mathbb{Z}_3$ contains subgroups isomorphic to free groups of any countable rank, as well as many different kinds of free products whose free factors are mixtures of free groups and cyclic groups of order 3.