Isomorphism in Quotients

Algebra Level 4

Let $G$ be a group. Let $H$ and $K$ be distinct normal subgroups of $G$ such that $H \simeq K$ .

True or False:

$\frac{G}{H} \simeq \frac{G}{K}$

False True

1 solution

Brian Moehring
Jun 14, 2018

For a class of easy counterexamples, consider $G=\mathbb{Z}$ and $H,K$ any two nontrivial subgroups.

For instance, $H=2\mathbb{Z}$ and $K=3\mathbb{Z}$ . Then $G/H \simeq \mathbb{Z}_2$ has two elements and $G/K \simeq \mathbb{Z}_3$ has three elements, so they're obviously not isomorphic.

Isomorphism in Quotients

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