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  "body": "\n## Statement\n\nLet $G$ be a [[group|G5S1M9]].\nLet $A$ and $B$ be [[subgroups|0X1JXT47GA]] of $G$, let $A^*$ be a [[normal subgroup|0CB0DX5BVV]] of $A$, and let $B^*$ be a normal subgroup of $B$.\nThen:\n\n1. $A^*(A \\cap B^*)$ is a normal subgroup of $A^*(A \\cap B)$;\n2. $B^*(A^* \\cap B)$ is a normal subgroup of $B^*(A \\cap B)$;\n3. there is an isomorphism of [[quotient groups|0KYX2N69J0]]\n$$\\frac{A^*(A \\cap B)}{A^*(A \\cap B^*)} \\cong \\frac{B^*(A \\cap B)}{B^*(A^* \\cap B)}.$$\n\n## Remarks\n\nLet $D = (A^* \\cap B)(A \\cap B^*)$.\nIt is a normal subgroup of $A \\cap B$, and the [[second isomorphism theorem|0PPP2J2DJV]] shows that the left quotient is isomorphic to $(A \\cap B)/D$.\nThe same argument with $A$ and $B$ exchanged shows that the right quotient is also isomorphic to $(A \\cap B)/D$.\nThe lemma is also called the butterfly lemma, after the shape of the diagram of the subgroups involved.\nFor $A^* = \\{e\\}$ and $B^* = \\{e\\}$ the lemma says only that $A \\cap B \\cong A \\cap B$; the content appears when $A^*$ and $B^*$ are large.\n\n## Explanations\n\n### L1 human\n\nThe lemma compares two pieces of a group, one cut out near $A$ and one cut out near $B$.\nIt says that the part of the first piece seen through $B$ is the same as the part of the second piece seen through $A$.\nIt is a technical tool, but it is exactly what makes two ways of cutting a group comparable.\n\n### L2 human\n\nThe lemma is the key step in the proof of the [[Schreier refinement theorem|QQAWHC9X60]], which in turn proves the [[Jordan–Hölder theorem|0E088ZP533]].\nIt is a consequence of the second isomorphism theorem alone.\nThe same statement and proof hold for modules over a ring, with sums in place of products of subgroups.\n",
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