Find $\sigma$ in $S_5$ that fulfills: $$\sigma(12)(34)=\sigma^{-1}(13)(45)$$ Can anyone help with that? i've tried to multiply by $\sigma^{-1}$ from the right side but it doesn't seem to lead anywhere
2026-03-30 13:17:09.1774876629
Symmetric groups
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You need $\sigma^2=(13)(45)(12)(34)=(12354)$ Clearly $\sigma$ must be a $5$-cycle also, so we get $\sigma=\sigma^6=(12354)^3=(15243)$