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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: | |||
* 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–Witt identity== | |||
'''Hall–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–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 .
- 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
Then .[1]
More generally, if , then if and , then .[2]
Proof and the Hall–Witt identity
Hall–Witt identity
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 .
See also
Notes
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References
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