What is the best way to prove that if a group is a subgroup of some other group? Or more precisely how to prove that they have common identity element?
2026-04-02 21:47:00.1775166420
proving the identity for subgroups.
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Let $e$ be the identity of $G$ and $e'$ be the identity of $H$.
Then $e'e'=e'$ in $H$.
This also holds in $G$ because the multiplication in $H$ is the restriction of the multiplication in $G$.
Multiplying by the inverse of $e'$ in $G$, we cancel $e'$ on both sides to get $e'=e$.