Let $n \in \mathbb{N}$ and let $S_n$ denote the permutation group on $n$ letters. I'm trying to figure out what kind of elements $\sigma, \delta \in S_n$ satisfy the following relations: $\sigma^2=e$, $\delta^4=e$, and $(\sigma\delta)^4=e$, where $e$ is the identity permutation. Can anyone provide a classification statement about elements of this form? Thanks!
a question about elements of permutation groups
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group-theory
finite-groups
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0Sorry, I h$a$dn't re$a$lised you were stipulating finite permutations. That said, every finite homomorphic image of the (2, 4, 4) triangle group is generated by some permutations of the form you want. Are you sure that there is a bound on the order of the finite homomorphic images of the (2, 4, 4) triangle group? Otherwise, what you are wanting to do will be exceptionally difficult (if not impossible?). On the other hand, if you prove that there are only a couple of types of permutations and so there are only finitely many permutation groups of this form then that's a nice result... – 2011-09-13