In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators. It is a consequence of the Hall–Witt identity.
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).
Let X, Y and Z be subgroups of a group G, and assume
More generally, if , then if and , then .
Proof and the 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 .
- ^ Isaacs, Lemma 8.27, p. 111
- ^ Isaacs, Corollary 8.28, p. 111