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!

  • $\begingroup$ $\sigma$ 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. $\endgroup$ – lhf Sep 12 '11 at 16:37
  • $\begingroup$ @Jack, thanks, I've deleted that comment and added another. $\endgroup$ – lhf Sep 12 '11 at 16:38
  • $\begingroup$ @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. $\endgroup$ – Arturo Magidin Sep 12 '11 at 16:41
  • $\begingroup$ What is the motivation for this question? $\endgroup$ – Arturo Magidin Sep 12 '11 at 16:46
  • $\begingroup$ @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? $\endgroup$ – Jack Schmidt Sep 12 '11 at 16:46

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