---
tag: 21JJ523V95
kind: statement
role: lemma
name: Zassenhaus lemma
msc: ["20E15", 20D30]
uses: [0X1JXT47GA, 0CB0DX5BVV, 0KYX2N69J0, 0PPP2J2DJV]
about: [0CB0DX5BVV, 0KYX2N69J0]
sources:
  - url: https://en.wikipedia.org/wiki/Zassenhaus_lemma
same_as:
  wikidata: Q3229348
  wikipedia: Zassenhaus lemma
---

## Statement

Let $G$ be a [[group|G5S1M9]].
Let $A$ and $B$ be [[subgroups|0X1JXT47GA]] of $G$, let $A^*$ be a [[normal subgroup|0CB0DX5BVV]] of $A$, and let $B^*$ be a normal subgroup of $B$.
Then:

1. $A^*(A \cap B^*)$ is a normal subgroup of $A^*(A \cap B)$;
2. $B^*(A^* \cap B)$ is a normal subgroup of $B^*(A \cap B)$;
3. there is an isomorphism of [[quotient groups|0KYX2N69J0]]
$$\frac{A^*(A \cap B)}{A^*(A \cap B^*)} \cong \frac{B^*(A \cap B)}{B^*(A^* \cap B)}.$$

## Remarks

Let $D = (A^* \cap B)(A \cap B^*)$.
It is a normal subgroup of $A \cap B$, and the [[second isomorphism theorem|0PPP2J2DJV]] shows that the left quotient is isomorphic to $(A \cap B)/D$.
The same argument with $A$ and $B$ exchanged shows that the right quotient is also isomorphic to $(A \cap B)/D$.
The lemma is also called the butterfly lemma, after the shape of the diagram of the subgroups involved.
For $A^* = \{e\}$ and $B^* = \{e\}$ the lemma says only that $A \cap B \cong A \cap B$; the content appears when $A^*$ and $B^*$ are large.

## Explanations

### L1 human

The lemma compares two pieces of a group, one cut out near $A$ and one cut out near $B$.
It says that the part of the first piece seen through $B$ is the same as the part of the second piece seen through $A$.
It is a technical tool, but it is exactly what makes two ways of cutting a group comparable.

### L2 human

The lemma is the key step in the proof of the [[Schreier refinement theorem|QQAWHC9X60]], which in turn proves the [[Jordan–Hölder theorem|0E088ZP533]].
It is a consequence of the second isomorphism theorem alone.
The same statement and proof hold for modules over a ring, with sums in place of products of subgroups.