Three subgroups lemma

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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[clarification needed] 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


Then .[1]

More generally, if , then if and , then .[2]

Proof and the Hall–Witt identity[edit]

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[edit]


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