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Problem 82 (Properties Of Cosets)

Let $G$ be a group, and let $H$ be a subgroup of $G$. Let $a,b \in G$. Prove the following:

  1. The function $f:H\to Ha$ defined by $f(h)=ha$ is a bijection.
  2. We have $|H|=|Ha|$.
  3. We know $b\in Ha$ if and only if $Hb=Ha$.

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