Q4

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

(A)

Show that $|aH| = |bH|.$

(B)

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

(C)

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

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