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In [[mathematics]], more specifically [[group theory]], the '''three subgroups lemma''' is a result concerning [[commutator]]s. It is a consequence of the [[Commutator#Identities|Hall–Witt identity]].


==Notation==
In that which follows, the following notation will be employed:


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* If ''H'' and ''K'' are subgroups of a group ''G'', the [[commutator]] of ''H'' and ''K'' will be denoted by [''H'',''K'']; 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 <math>x^{y}</math>.
* If ''H'' is a subgroup of a group ''G'', then the [[Centralizer and normalizer|centralizer]] of ''H'' in ''G'' will be denoted by '''C'''<sub>G</sub>(''H'').
 
== Statement ==
Let ''X'', ''Y'' and ''Z'' be subgroups of a group ''G'', and assume
 
:<math>[X,Y,Z]=1</math> and <math>[Y,Z,X]=1</math>
 
Then <math>[Z,X,Y]=1</math>.<ref>Isaacs, Lemma 8.27, p. 111</ref>
 
More generally, if <math>N\triangleleft G</math>, then if <math>[X,Y,Z]\subseteq N</math> and <math>[Y,Z,X]\subseteq N</math>, then <math>[Z,X,Y]\subseteq N</math>.<ref>Isaacs, Corollary 8.28, p. 111</ref>
 
==Proof and the Hall&ndash;Witt identity==
 
'''Hall&ndash;Witt identity'''
 
If <math>x,y,z\in G</math>, then
 
<math>[x, y^{-1}, z]^y\cdot[y, z^{-1}, x]^z\cdot[z, x^{-1}, y]^x = 1</math>
 
'''Proof of the Three subgroups lemma'''
 
Let <math>x\in X</math>, <math>y\in Y</math>, and <math>z\in Z</math>. Then <math>[x,y^{-1},z]=1=[y,z^{-1},x]</math>, and by the Hall&ndash;Witt identity above, it follows that <math>[z,x^{-1},y]^{x}=1</math> and so <math>[z,x^{-1},y]=1</math>. Therefore, <math>[z,x^{-1}]\subseteq \bold{C}_G(Y)</math> for all <math>z\in Z</math> and <math>x\in X</math>. Since these elements generate <math>[Z,X]</math>, we conclude that <math>[Z,X]\subseteq \bold{C}_G(Y)</math> and hence <math>[Z,X,Y]=1</math>.
 
==See also==
*[[Commutator#Group_theory|Commutator]]
*[[Lower central series]]
*[[Grün's lemma]]
 
==Notes==
{{reflist|2}}
 
==References==
* {{cite book
| author = I. Martin Isaacs
| year = 1993
| title = Algebra, a graduate course
| edition = 1st edition
| publisher = Brooks/Cole Publishing Company
| isbn = 0-534-19002-2
}}
 
[[Category:Theorems in group theory]]
[[Category:Articles containing proofs]]
[[Category:Lemmas]]

Revision as of 13:00, 28 January 2014

In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators. It is a consequence of the Hall–Witt identity.

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 will be denoted by [H,K]; 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 xy.
  • If H is a subgroup of a group G, then the centralizer of H in G will be denoted by CG(H).

Statement

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

[X,Y,Z]=1 and [Y,Z,X]=1

Then [Z,X,Y]=1.[1]

More generally, if NG, then if [X,Y,Z]N and [Y,Z,X]N, then [Z,X,Y]N.[2]

Proof and the Hall–Witt identity

Hall–Witt identity

If x,y,zG, then

[x,y1,z]y[y,z1,x]z[z,x1,y]x=1

Proof of the Three subgroups lemma

Let xX, yY, and zZ. Then [x,y1,z]=1=[y,z1,x], and by the Hall–Witt identity above, it follows that [z,x1,y]x=1 and so [z,x1,y]=1. Therefore, [z,x1]CG(Y) for all zZ and xX. Since these elements generate [Z,X], we conclude that [Z,X]CG(Y) and hence [Z,X,Y]=1.

See also

Notes

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