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Let $G$ be a finite group and $H$ be a subgroup of $G$. For $a \in G$, define $aH=\left\{ah \mid h \in H\right\}$.

  1. Show that $|aH| = |H|$

  2. Show that for every pair of elements $a, b \in G$, either $aH = bH$ or $aH$ and $bH$ are disjoint

  3. Use the above to argue that the order of $H$ must divide the order of $G$

 

asked in Set Theory & Algebra by Veteran (69k points) | 236 views

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