Inverses in Subgroup

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Theorem

Let $G$ be a group.

Let $H$ be a subgroup of $G$.


Then for each $h \in H$, the inverse of $h$ in $H$ is the same as the inverse of $h$ in $G$.


Proof

Let $h \in H$.

Let:

$h'$ be the inverse of $h$ in $H$
$h^{-1}$ be the inverse of $h$ in $G$.


From Identity of Subgroup, $h \circ h' = e$.

From Inverse in Group is Unique, it follows that $h' = h^{-1}$.

$\blacksquare$


Sources