- #1
NoName3
- 25
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Let $H$ be be a subgroup of a group $G$. Let $g'$ and $g$ elements of $G$. Prove that the following are equivalent: $(a)$ $g' \in Hg$, $(b)$ $g'g^{-1} \in H$, and $(c)$ $Hg = Hg'$.
$g' \in Hg$ means $g' = hg$ for some $g \in G$ and $h \in H$. And $g' = hg \implies g'g^{-1} = hgg^{-1} = h$. But $h \in H$ so $g'g^{-1} \in H$.
So $(a) \implies (b)$. I can't progress. For a moment I thought I had it, then I lost it!
$g' \in Hg$ means $g' = hg$ for some $g \in G$ and $h \in H$. And $g' = hg \implies g'g^{-1} = hgg^{-1} = h$. But $h \in H$ so $g'g^{-1} \in H$.
So $(a) \implies (b)$. I can't progress. For a moment I thought I had it, then I lost it!
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