# Three subgroups lemma

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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.

## Notation

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 ${\displaystyle x^{y}}$.
• If H is a subgroup of a group G, then the centralizer of H in G will be denoted by CG(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.

## Statement

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

${\displaystyle [X,Y,Z]=1}$ and ${\displaystyle [Y,Z,X]=1}$

Then ${\displaystyle [Z,X,Y]=1}$. [1]

More generally, if ${\displaystyle N\triangleleft G}$, then if ${\displaystyle [X,Y,Z]\subseteq N}$ and ${\displaystyle [Y,Z,X]\subseteq N}$, then ${\displaystyle [Z,X,Y]\subseteq N}$. [2]

## Proof and the Hall–Witt identity

HallWitt identity

If ${\displaystyle x,y,z\in G}$, then

${\displaystyle [x,y^{-1},z]^{y}\cdot [y,z^{-1},x]^{z}\cdot [z,x^{-1},y]^{x}=1.}$

Proof of the three subgroups lemma

Let ${\displaystyle x\in X}$, ${\displaystyle y\in Y}$, and ${\displaystyle z\in Z}$. Then ${\displaystyle [x,y^{-1},z]=1=[y,z^{-1},x]}$, and by the HallWitt identity above, it follows that ${\displaystyle [z,x^{-1},y]^{x}=1}$ and so ${\displaystyle [z,x^{-1},y]=1}$. Therefore, ${\displaystyle [z,x^{-1}]\subseteq \mathbf {C} _{G}(Y)}$ for all ${\displaystyle z\in Z}$ and ${\displaystyle x\in X}$. Since these elements generate ${\displaystyle [Z,X]}$, we conclude that ${\displaystyle [Z,X]\subseteq \mathbf {C} _{G}(Y)}$ and hence ${\displaystyle [Z,X,Y]=1}$.

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.

## Notes

1. Isaacs, Lemma 8.27, p. 111
2. Isaacs, Corollary 8.28, p. 111

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## References

• 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.