Consider the group that is the free product of two copies of $\mathbb{Z}$, that is $G=\mathbb{Z}_2 * \mathbb{Z}_2=\left\langle a,b\;\vert \;a^2=b^2=e \right\rangle$. I'm trying to find all of its proper and non trivial subgroups. I can find at least one subgroup, ie I think that there is a subgroup of $G$ generated by $ab$, which is isomorphic to $\mathbb{Z}$. But that's about all the obvious one I can see straight away and I'm not sure how to go about finding the rest.
The reason I'm asking this is because I'm trying to find the all of the connected covering spaces of $\mathbb{R}P^2\vee\mathbb{R}P^2$. If I'm not mistaken, this space has $G$ as its fundamental group, hence identifying subgroups of $\pi_1(G)$ would allow me to classify the covering spaces (right?).
If anyone can provide some insight I'd appreciate that, thank you!
EDIT: I forgot to write it down but of course by definition $H*H'$ contains both $H$ and $H'$ as subgroups so here we also have $\mathbb{Z}_2$ as a subgroup. $G$ even has two such subgroups, the one generated by $a$ and the one generated by $b$ (right?).
Best Answer
Let $t = ab$. Then $G = \langle a,t \mid a^2=1, a^{-1}ta=t^{-1} \rangle$ is the infinite dihedral group.
The subgroups are as follows (apologies in advance for any mistakes).
$\mathbf{Type\ 1.}$ The trivial subgroup.
$\mathbf{Type\ 2.}$ Infinite cyclic subgroups $C_k = \langle t^k \rangle$, one for each integer $k>0$. Note that $C_k$ has index $k$ in $\langle t \rangle$.
$\mathbf{Type\ 3.}$ Reflection subgroups $R_k= \langle t^ka \rangle$, one for each $k \in {\mathbb Z}$. They have order $2$.
$\mathbf{Type\ 4.}$ Infinite dihedral subgroups $D_{ki} = \langle t^k,t^ia \rangle$, one for each $k>0$ and $0\le i < k$. These have finite index $k$ in $G$.