Three subgroups lemma

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


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


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 HallWitt identity

HallWitt identity

If , then

Proof of the three subgroups lemma

Let , , and . Then , and by the HallWitt identity above, it follows that and so . Therefore, for all and . Since these elements generate , we conclude that and hence .

See also


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


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