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

**Mathematics** includes the study of such topics as quantity, structure, space, and change.

In mathematics and abstract algebra, **group theory** studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

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 that which 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*).

In mathematics, especially group theory, the **centralizer** of a subset *S* of a group *G* is the set of elements of *G* that commute with each element of *S*, and the **normalizer** of *S* are elements that satisfy a weaker condition. The centralizer and normalizer of *S* are subgroups of *G*, and can provide insight into the structure of *G*.

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

- and

Then .^{ [1] }

More generally, if , then 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 the **Jacobi identity** is a property of a binary operation which describes how the order of evaluation 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. The cross product and the Lie bracket operation both satisfy the Jacobi identity.

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 **group action** is a formal way of interpreting the manner in which the elements of a group correspond to transformations of some space in a way that preserves the structure of that space. Common examples of spaces that groups act on are sets, vector spaces, and topological spaces. Actions of groups on vector spaces are called representations of the group.

In mathematics, a **Lie algebra** is a vector space together with a non-associative, alternating bilinear map , called the Lie bracket, satisfying the Jacobi identity.

In mathematics, a **ring** is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices and functions.

An **exact sequence** is a concept in mathematics, especially in group theory, ring and module theory, homological algebra, as well as in differential geometry. An exact sequence is a sequence, either finite or infinite, of objects and morphisms between them 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. Equivalently, it is the group of *n*×*n* orthogonal matrices, where the group operation is given by matrix multiplication; an orthogonal matrix is a real matrix whose inverse equals its transpose.

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.

In mathematics, specifically group theory, the **index** of a subgroup *H* in a group *G* is the "relative size" of *H* in *G*: equivalently, the number of "copies" (cosets) of *H* that fill up *G*. For example, if *H* has index 2 in *G*, then intuitively half of the elements of *G* lie in *H*. The index of *H* in *G* is usually denoted |*G* : *H*| or [*G* : *H*] or (*G*:*H*).

In mathematics, the **adjoint representation** of a Lie group *G* is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if *G* is GL(*n*), its Lie algebra is the vector space of all *n*-by-*n* matrices. So in this example, the adjoint representation is the vector space of *n*-by-*n* matrices , and any element *g* in GL(*n*) acts as a linear transformation of this vector space given by conjugation: .

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 an 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, a **universal enveloping algebra** is the most general algebra that contains all representations of a 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 the area of abstract algebra known as group theory, a group is said to be **perfect** if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial 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 while *K* acts on *G* by right multiplication. For each *x* in *G*, the **( H, K)-double coset of x** is the set

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, the **ping-pong lemma**, or **table-tennis lemma**, is any of several mathematical statements that ensure that several elements in a group acting on a set freely generates a free subgroup of that group.

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 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.

In the theory of Lie groups, the **exponential map** is a map from the Lie algebra of a Lie group to the group, which allows one to recapture the local group structure from the Lie algebra. The existence of the exponential map is one of the primary reasons that Lie algebras are a useful tool for studying Lie groups.

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

**I. Martin "Marty" Isaacs** is a group theorist and representation theorist and professor emeritus of mathematics at the University of Wisconsin–Madison. He currently lives in Berkeley, California and is an occasional participant on MathOverflow.

The **International Standard Book Number** (**ISBN**) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.

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