Zassenhaus lemma

The Zassenhaus lemma states that two sections formed from subgroups of a group are isomorphic. It is also called the butterfly lemma.

Group: A set equipped with an associative binary operation, an identity element, and inverses for every element. The lemma applies to subgroup configurations inside a single group.
Intersection of subgroups: The set of elements common to two or more subgroups of a group. The lemma uses intersections to define the normal subgroups in its sections.
Schreier refinement theorem: A theorem stating that any two normal series of a group have equivalent refinements. Repeated applications of the lemma match the factors in refined subgroup series.
Hans Zassenhaus: A German-American mathematician known for work in group theory and algebra. The lemma bears Zassenhaus's name.
Subgroup: A subset of a group that is itself a group under the inherited operation. The lemma begins with two subgroups and their intersections.
Product of subgroups: The set of products of elements drawn from two subgroups of the same group. Products with the opposite subgroup form the numerators of the two sections.
Jordan–Hölder theorem: A theorem stating that composition factors of a finite group are unique up to order and isomorphism. The refinement argument uses the lemma to compare factors from different composition series.
Butterfly lemma: An alternate name for the Zassenhaus lemma, inspired by the shape of its subgroup diagram. The diagram's paired wings make the theorem's subgroup arrangement visually memorable.
Normal subgroup: A subgroup invariant under conjugation by every element of its ambient group. The quotient groups in the lemma require normal subgroups.
Conjugate subgroup: A subgroup obtained by conjugating every element of a subgroup by one fixed group element. Conjugation invariance is the test that makes the section denominators normal.

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