## Group generated by x, y where |x| = p, a prime, and |y| = 2 isomorphic to dihedral group if the group generated by x, y is of order 2p [closed]

How can I prove this?

Group generated by x, y where |x| = p, a prime, and |y| = 2 isomorphic to dihedral group if the group generated by x, y is of order 2p

This is a part of a larger proof.

I proved that all element of the group can be written uniquely x^my^n. (Unique meaning if x^my^n = g = x^m' y^n', then m = m' mod p and n = n' mod 2)

I was going to prove the following map was a group isomorphism.

f : G -> D_{2p} by g = x^my^n |-> r^mf^n

where r is a rotation and f is a flip in the dihedral group.

I can do everything if I can show (xy)^2 = e. But I am not sure how to do that, ive had a few failed attempts.

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It is not true. $PSL(2,\mathbb{Z})$ is generated by an element of order 3 and an element of order 2. – Mark Sapir Nov 13 2011 at 22:24
If I restrict the order of the whole group to 2p, does it work? – Setraced Nov 13 2011 at 22:26
Groups of order $pq$, $p,q$ are primes, are well known. That is not a research question. – Mark Sapir Nov 13 2011 at 22:37
Try math.stackexchange. – Gerry Myerson Nov 13 2011 at 22:41
This better asked on math.stackexchange. As a push though, what possibilities are there for yxy? for yyxyy? Gerhard "Ask Me About System Design" Paseman, 2011.11.13 – Gerhard Paseman Nov 13 2011 at 22:44