= Zassenhaus butterfly lemma
{c}
Let $A'\trianglelefteq A$ and $B'\trianglelefteq B$ be <subgroups> of one <group>. Put $C=A\cap B$ and $D=(A'\cap B)(A\cap B')$. Then $D\trianglelefteq C$ and the <first isomorphism theorem> identifies both
$$
\frac{A'C}{A'(A\cap B')}\quad\text{and}\quad\frac{B'C}{B'(B\cap A')}
$$
with $C/D$. Indeed $C$ maps onto either displayed <quotient group>, and its kernel in either case is $D$. This gives the matching factors used in the <Schreier refinement theorem>.
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