Let $G_1$ be a finite group with a subgroup $G_2$ and a normal subgroup $H$. Suppose that $G_1/H$ is isomorphic to a subgroup of $G_2$. Do we have $G_1/H$ is isomorphic to $G_2$?


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Definitely not.

Take $G_1=\mathbb{Z}_2\oplus \mathbb{Z}_4$, let $G_2$ be the $\mathbb{Z}_4$ summand, and let $$H:=\mathbb{Z}_2\oplus 2\mathbb{Z}_4=\{(0,0),(0,2),(1,0),(1,2)\}.$$ Then $G_1/H$ is a group of order $2$, so it's isomorphic to a subgroup of $\mathbb{Z}_4\simeq G_2$. But of course it's not isomorphic to $\mathbb{Z}_4$.

Direct sums are a great place to start looking for examples in situations like these (quotients of subgroups, etc).


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