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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!

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    $\sigma$ is a product of disjoint transpositions. $\delta$ and $\sigma\delta$ are a product of disjoint transpositions and 4-cycles with at least one 4-cycle.2011-09-12
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    @Jack, thanks, I've deleted that comment and added another.2011-09-12
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    @lhf: Not necessarily: $\delta^4=e$ means the order of $\delta$ is either $1$, $2$, or $4$, so you could have $\delta$ be a product of transpositions. $\sigma=(1,2)$ and $\delta=(3,4)$ satisfy all three conditions.2011-09-12
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    What is the motivation for this question?2011-09-12
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    @dan, I think looking at the cycle decomposition is the right idea, but I am nervous as to how much this can actually tell you, as there appears to be quite some variety in the group generated by sigma and delta. Do you care about that group structure, or just how the cycles of delta and sigma relate?2011-09-12
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    @Jack: I don't really care about the corresponding group, I'd just like to know what the permutations look like, so (as you say) how the cycles of delta and sigma relate.2011-09-12
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    @Arturo: It would take awhile to give the motivation. Basically, if I can understand how the cycles relate then I can probably build a counterexample to a question posed to me.2011-09-12
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    @dan: It seems hard to me; even restricting to $\delta^2=e$, there are several possibilities for *each* block of $4$ elements permuted by $\sigma\delta$; you *may* try (and I don't know for sure this is true) that if $\sigma\delta$ maps $\{a,b,c,d\}$ to itself, then $\sigma$ and $\delta$ must also map it to itself. If this holds, then you'll be able to at least break up $n$ into tractable blocks.2011-09-12
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    @dan: It doesn't work. You can have something like $\delta=(a,b)(c,d)(f)(g)$, and $\sigma=(a,g)(b,f)(c)(d)$. Then $\sigma\delta=(a,f,b,g)(c,d)$. I think this will prove very difficult to characterize.2011-09-12
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    Such permutations need not be finite. This is because the group with the relations you gave is infinite (look up Euclidean triangle group), and then apply Cayley's Theorem.2011-09-12
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    @Swlabr: I'm not sure what you mean; I'm assuming that the permutations are elements of a finite permutation group. I'm aware that the group $\langle a,b | a^2=b^4=(ab)^4=1 \rangle$ is infinite, but I'm not studying that group. I just want to know if there are any relationships between the cycle structures of a sigma and a delta (elements of a finite permutation group) as described above.2011-09-12
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    Sorry, I hadn't realised 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

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