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Let $G_1, G_2$ be groups, and let $N_1\trianglelefteq G_1, N_2\trianglelefteq G_2$ such that $G_1\cong G_2, N_1\cong N_2$. Prove or disprove $G_1/N_1\cong G_2/N_2$.

Let $\varphi:G_1\to G_2, \psi:N_1\to N_2$ be isomorphisms.

I think that if this isn't true then I have to find an example such that $\varphi[N_1]\ne N_2$. I didn't find any disproofing.

J. Doe
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1 Answers1

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A counterexample: $G_1=G_2=\mathbb Z$, $N_1 = \mathbb Z$, and $N_2 = 2 \mathbb Z$ which is isomorphic to $\mathbb Z$. Then $G_1 / N_1$ is trivial and $G_2 / N_2$ is cyclic of order 2.

Lee Mosher
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