Let $H=\{e,(13)\}$ be a subgroup of $S_3$. Find element $a,b \in S_3$ where $bh_2ah_1 \in aH$ but $bH\ne H$. $h_1$ and $h_2$ are elements in $H$.
My friend thinks that it is (123) and (132), but I cant see how he gets there. Could someone help me?
2026-03-28 01:14:35.1774660475
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My friend was right. If $a=(123)$ and $b=(132)$, with $h_1=e$ and $h_2=(13)$, then $aH=${$(123),(23)$}. Also $bh_2ah_1$=$(23) \in aH$. And bH={$(132),(12)$}$\ne H$. So left and right make a difference.