Third isomorphism theorem

This article gives the statement, and possibly proof, of a basic fact in group theory.
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Statement

Suppose $G$ is a group, and $H$ and $K$ are normal subgroups of $G$, such that $H\le K$. Then we have the following natural isomorphism:

$(G/H)/(K/H)\cong G/K$

Where the isomorphism sends a coset $Hg$ in $G$ to the coset $Kg$ in $G$.

Note that this statement makes sense at the level of a group isomorphism only when both $H$ and $K$ are normal in $G$. Otherwise, the statement is still true at the level of sets, but we cannot make sense of it as a group isomorphism.

Related facts

• First isomorphism theorem
• Second isomorphism theorem