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Problem 84 (Properties Of Cosets Part Two)

Let $H$ be a subgroup of $G$. Let $a,b\in G$. Prove the following facts about cosets.

  1. We have $Ha=Hb$ if and only if $ab^{-1}\in H$.
  2. We must have $Ha=Hb$ or $Ha\cap Hb=\emptyset$.
  3. We have $Ha=aH$ if and only if $aHa^{-1}=H$.

We've already seen all of these properties before when studying modular arithmetic. Let $G=\mathbb{Z}$ and let $H=n\mathbb{Z}$ for some $n\in\mathbb{N}$. The cosets of $H$ are $r+n\mathbb{Z}$ for $0\leq r<n$.

  • The first property above states that $a+n\mathbb{Z}=b+n\mathbb{Z}$ if and only if $a-b\in n\mathbb{Z}$. Wait, this just says two numbers have the same remainder if and only if their difference is a multiple of $n$.
  • The second property above states that either $a$ and $b$ have the same remainder, or they don't. It basically states that remainders are unique.
  • The third property is trivial for $\mathbb{Z}$ because $\mathbb{Z}$ is an Abelian group and we always know $a+n\mathbb{Z}=n\mathbb{Z}+a$.

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