Group theory and generalizations Finite groups Lemma

Zassenhaus lemma

Builds on Subgroup Normal subgroup Quotient group Second isomorphism theorem

Statement

Let $G$ be a group. Let $A$ and $B$ be subgroups of $G$ , let $A^*$ be a normal subgroup of $A$ , and let $B^*$ be a normal subgroup of $B$ . Then:

$A^*(A \cap B^*)$ is a normal subgroup of $A^*(A \cap B)$ ;
$B^*(A^* \cap B)$ is a normal subgroup of $B^*(A \cap B)$ ;
there is an isomorphism of quotient groups $\frac{A^*(A \cap B)}{A^*(A \cap B^*)} \cong \frac{B^*(A \cap B)}{B^*(A^* \cap B)}.$

Remarks

Let $D = (A^* \cap B)(A \cap B^*)$ . It is a normal subgroup of $A \cap B$ , and the second isomorphism theorem 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.

Going deeper

For a student

The lemma is the key step in the proof of the Schreier refinement theorem, which in turn proves the Jordan–Hölder theorem. 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.

Reading path

To understand this page from the ground up, read these 35 pages in order. The foundations come first. The path takes the proofs Proof that the kernel is a normal subgroup, Proof that a homomorphism is injective iff its kernel is trivial, Proof of the first isomorphism theorem, Proof of the uniqueness of inverses, Proof of the cancellation laws.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Binary operation definition
Group concept
Normal subgroup concept
Quotient group concept
Subgroup concept
Identity element definition
Conjugate elements definition
Group homomorphism definition
Subgroup definition
Inverse element definition
Conjugate subgroup definition
Group isomorphism definition
Kernel of a homomorphism definition
Image of a homomorphism definition
Left coset definition
Product of two subgroups definition
Right coset definition
Proof of the cancellation laws proof
Proof of the uniqueness of inverses proof
Normal subgroup definition
Cancellation laws statement
Uniqueness of inverses statement
Product of a subgroup and a normal subgroup is a subgroup statement
Normality via left and right cosets statement
Homomorphisms preserve identity and inverses statement
Quotient group definition
Proof that a homomorphism is injective iff its kernel is trivial proof
Proof that the kernel is a normal subgroup proof
Universal property of the quotient group statement
A homomorphism is injective iff its kernel is trivial statement
The kernel is a normal subgroup statement
Proof of the first isomorphism theorem proof
First isomorphism theorem statement
Second isomorphism theorem statement
Zassenhaus lemma statement

Where this leads

1 page builds on this one.

Schreier refinement theorem

Tag	`21JJ523V95`
URI	`m2://21JJ523V95`
Kind	statement, lemma
Writing	W0 stub
MSC	20E15, 20D30
Status	This copy of the site has no build data. Run `m2 status 21JJ523V95`.
About	Normal subgroup, Quotient group
Uses	Subgroup, Normal subgroup, Quotient group, Second isomorphism theorem
Same as	Wikidata Q3229348 Wikipedia Zassenhaus lemma
Changed	2 November 2026

Zassenhaus lemma

Statement

Remarks

Going deeper

For a student

Reading path

Where this leads

Further reading

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