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We define $D_{\infty} := \langle r, s \ | \ s^2 = e, srs = r^{-1} \rangle = \lbrace ..., r^{-2}, r^{-1}, r, r^2, ..., e, ..., r^{-2}s, r^{-1}s, s, rs, r^2s, ... \rbrace$. I am trying to find a subgroup $H < D_{\infty}$ such that $H \neq D_{\infty}$ but $H \simeq D_{\infty}$. I have been trying some things with the group $H := \lbrace ..., r^{-4}, r^{-2}, r^2, r^4, ..., e, ..., r^{-4}s, r^{-2}s, s, r^2s, r^4s ... \rbrace$. However, I have not been able to find an isomorphism between this group and $D_{\infty}$. Any thoughts on a possible isomorphism, or on the question if this is even the right $H$, would be welcome.

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Since $sr^2s = (r^2)^{-1}$, the assignment $s \mapsto s$ and $r \mapsto r^2$ defines a homomorphism $D_\infty \to H$, and it is not difficult to see that this homomorphism is surjective and injective.

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