Siena Duran

2022-02-12

Let H be a subgroup of a group G and $a,b\in G$. Then $a\in bH$ if and only if
1) $ba-1\in H$
2) $ba\in H$
3) $b-1a\in H$
4) None of these

Johnathan Carpenter

Given:
H be a subgroup of a group G.
i.e. $H\subseteq G$.
and $a,b\in G$ If $a\in bH$ (1)
$b\in bH$
$⇒{b}^{1}\in bH\left(\therefore a\in G⇒{a}^{1}\in G\right)$ (2)
Now from eq-n (1) and (2)

then ${b}^{1}a\in {b}^{1}\left(bH\right)$
${b}^{1}a\in \left({b}^{1}b\right)H$
${b}^{1}a\in H$

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