In $S_5$, we have $aba^{-1}=b^2, b=(12345)$, find $a$.
I have tried different ways to substitute/rearrange, but none of them worked.
2026-04-02 04:30:28.1775104228
In $S_5$, we have $aba^{-1}=b^2$, $b=(12345)$, find $a$.
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Use the fact that for any cycle $(abc...)$ and permutation $\sigma$, $\sigma(abc...)\sigma^{-1}=(\sigma(a)\sigma(b)...)$. In your case, $b^2=(13524)$, so $a(1)=1$, $a(2)=3$, $a(3)=5$, $a(4)=2$, $a(5)=4$, and putting this all together in cycle notation: $a=(2354)$.