The Frattini subgroup of $D_{\infty}$ [closed]

Question

Please hint me. $\phi(D_{\infty})?$ $\phi(G)$ is Frattini subgroup of $G$, intersection of all the maximal subgroup of $G$ and $D_{\infty}=<x,y|x^2=y^2=1>$.

Answer 1

Put $a := xy$ and $b := y$. Then we have ${\rm D}_\infty = \langle a, b \rangle$. Clearly $a$ generates a subgroup of index 2, which is thus maximal. Also, given a prime $p$, the subgroup $G_p := \langle a^p, b \rangle$ has index $p$ in ${\rm D}_\infty$, and is thus maximal as well. The intersection of the groups $G_p$ is $\langle b \rangle$, which intersects trivially with $\langle a \rangle$. Therefore the Frattini subgroup is trivial.