Goursat's Lemma

Lemma

Let $G,G^{\prime }$ be groups, and let $H$ be a subgroup of $G\times G^{\prime }$ such that the two projections $p_1:H\to G$ and $p_2:H\to G^{\prime }$ are surjective (i.e., $H$ is a subdirect product of $G$ and $G^{\prime })$. Let $N$ be the kernel of $p_2$ and $N^{\prime }$ the kernel of $p_1.$ One can identify $N$ as a normal subgroup of $G$, and $N^{\prime }$ as a normal subgroup of $G^{\prime }$. Then the image of $H$ in $G/N\times G^{\prime }/N^{\prime }$ is the graph of an isomorphism $G/N\cong G^{\prime }/N^{\prime }$. One then obtains a bijection between:

• Subgroups of $G\times G^{\prime }$ which project onto both factors, and
• Triples $(N,N^{\prime },f)$ with $N$ normal in $G,N^{\prime }$ normal in $G^{\prime }$ and $f$ isomorphism of $G/N$ onto $G^{\prime }/N^{\prime }$.