In mathematics, more specifically group theory, the **three subgroups lemma** is a result concerning commutators. It is a consequence of Philip Hall and Ernst Witt's eponymous identity.

In what follows, the following notation will be employed:

- If
*H*and*K*are subgroups of a group*G*, the commutator of*H*and*K*, denoted by [*H*,*K*], is defined as the subgroup of*G*generated by commutators between elements in the two subgroups. If*L*is a third subgroup, the convention that [*H*,*K*,*L*] = [[*H*,*K*],*L*] will be followed. - If
*x*and*y*are elements of a group*G*, the conjugate of*x*by*y*will be denoted by . - If
*H*is a subgroup of a group*G*, then the centralizer of*H*in*G*will be denoted by**C**_{G}(*H*).

Let *X*, *Y* and *Z* be subgroups of a group *G*, and assume

- and

Then .^{ [1] }

More generally, for a normal subgroup of , if and , then .^{ [2] }

**Hall–Witt identity**

If , then

**Proof of the three subgroups lemma**

Let , , and . Then , and by the Hall–Witt identity above, it follows that and so . Therefore, for all and . Since these elements generate , we conclude that and hence .

In mathematics, an **abelian group**, also called a **commutative group**, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.

In mathematics, the **commutator** gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.

In mathematics, more specifically in abstract algebra, the **commutator subgroup** or **derived subgroup** of a group is the subgroup generated by all the commutators of the group.

In mathematics, a **Lie algebra** is a vector space together with an operation called the **Lie bracket**, an alternating bilinear map , that satisfies the Jacobi identity. The Lie bracket of two vectors and is denoted . The vector space together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative.

In mathematics, **rings** are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a *ring* is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

A **mathematical symbol** is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.

An **exact sequence** is a sequence of morphisms between objects such that the image of one morphism equals the kernel of the next.

In mathematics, the **orthogonal group** in dimension *n*, denoted O(*n*), is the group of distance-preserving transformations of a Euclidean space of dimension *n* that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the **general orthogonal group**, by analogy with the general linear group. Equivalently, it is the group of *n*×*n* orthogonal matrices, where the group operation is given by matrix multiplication. The orthogonal group is an algebraic group and a Lie group. It is compact.

In mathematics, the **unitary group** of degree *n*, denoted U(*n*), is the group of *n* × *n* unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL(*n*, **C**). **Hyperorthogonal group** is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group.

A group is a set together with an associative operation which admits an identity element and such that every element has an inverse.

In mathematics, the **Jacobi identity** is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associative property, any order of evaluation gives the same result. The identity is named after the German mathematician Carl Gustav Jakob Jacobi.

In mathematics, specifically group theory, the **free product** is an operation that takes two groups *G* and *H* and constructs a new group *G* ∗ *H*. The result contains both *G* and *H* as subgroups, is generated by the elements of these subgroups, and is the “universal” group having these properties, in the sense that any two homomorphisms from *G* and *H* into a group *K* factor uniquely through a homomorphism from *G* ∗ *H* to *K*. Unless one of the groups *G* and *H* is trivial, the free product is always infinite. The construction of a free product is similar in spirit to the construction of a free group.

In mathematics, the **universal enveloping algebra** of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.

In mathematics, the **Heisenberg group**, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form

In mathematics, more specifically in group theory, a group is said to be **perfect** if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients. In symbols, a perfect group is one such that *G*^{(1)} = *G*, or equivalently one such that *G*^{ab} = {1}.

In group theory, a field of mathematics, a **double coset** is a collection of group elements which are equivalent under the symmetries coming from two subgroups. More precisely, let *G* be a group, and let *H* and *K* be subgroups. Let *H* act on *G* by left multiplication and let *K* act on *G* by right multiplication. For each *x* in *G*, the **( H, K)-double coset of x** is the set

In mathematics, specifically in group theory, the **direct product** is an operation that takes two groups *G* and *H* and constructs a new group, usually denoted *G* × *H*. This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.

In mathematics, especially in the fields of group theory and Lie theory, a **central series** is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, this is an explicit expression that the group is a nilpotent group, and for matrix rings, this is an explicit expression that in some basis the matrix ring consists entirely of upper triangular matrices with constant diagonal.

In mathematics, in the areas of group theory and combinatorics, **Hall words** provide a unique monoid factorisation of the free monoid. They are also totally ordered, and thus provide a total order on the monoid. This is analogous to the better-known case of Lyndon words; in fact, the Lyndon words are a special case, and almost all properties possessed by Lyndon words carry over to Hall words. Hall words are in one-to-one correspondence with **Hall trees**. These are binary trees; taken together, they form the **Hall set**. This set is a particular totally ordered subset of a free non-associative algebra, that is, a free magma. In this form, the Hall trees provide a basis for free Lie algebras, and can be used to perform the commutations required by the Poincaré–Birkhoff–Witt theorem used in the construction of a universal enveloping algebra. As such, this generalizes the same process when done with the Lyndon words. Hall trees can also be used to give a total order to the elements of a group, via the commutator collecting process, which is a special case of the general construction given below. It can be shown that **Lazard sets** coincide with Hall sets.

In mathematics, the **complexification** or **universal complexification** of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that every continuous homomorphism of the original group into another complex Lie group extends compatibly to a complex analytic homomorphism between the complex Lie groups. The complexification, which always exists, is unique up to unique isomorphism. Its Lie algebra is a quotient of the complexification of the Lie algebra of the original group. They are isomorphic if the original group has a quotient by a discrete normal subgroup which is linear.

- I. Martin Isaacs (1993).
*Algebra, a graduate course*(1st ed.). Brooks/Cole Publishing Company. ISBN 0-534-19002-2.

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