Three subgroups lemma

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.

Notation

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 (group theory), conjugate of ''x'' by ''y'' will be denoted by $x^$.
* If ''H'' is a subgroup of a group ''G'', then the centralizer of ''H'' in ''G'' will be denoted by C_G(''H'').

Statement

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

: ,Y,Z1 and ,Z,X1.

Then ,X,Y1.

More generally, for a normal subgroup $N$ of $G$, if ,Y,Zsubseteq N and ,Z,Xsubseteq N, then ,X,Ysubseteq N.Isaacs, Corollary 8.28, p. 111

Proof and the Hall–Witt identity

Hall–Witt identity

If $x,y,z\in G$, then

: , y^, zy\cdot , z^, xz\cdot , x^, yx = 1.

Proof of the three subgroups lemma

Let $x\in X$, $y\in Y$, and $z\in Z$. Then ,y^,z1= ,z^,x/math>, and by the Hall–Witt identity above, it follows that ,x^,y=1 and so ,x^,y1. Therefore, ,x^in \mathbf_G(Y) for all $z\in Z$ and $x\in X$. Since these elements generate ,X/math>, we conclude that ,Xsubseteq \mathbf_G(Y) and hence ,X,Y1.

See also

*Commutator
*Lower central series
* Grün's lemma
*Jacobi identity

Notes

References

* {{cite book
, author = I. Martin Isaacs
, author-link = Martin Isaacs
, year = 1993
, title = Algebra, a graduate course
, edition = 1st
, publisher = Brooks/Cole Publishing Company
, isbn = 0-534-19002-2

Lemmas in group theory
Articles containing proofs