Three subgroups lemma
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
- and
Then .[1]
More generally, if , then if and , then .[2]
Proof and the Hall–Witt identity
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 .
See also
Notes
References
- I. Martin Isaacs (1993). Algebra, a graduate course (1st ed.). Brooks/Cole Publishing Company. ISBN 0-534-19002-2.
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