# 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.

## 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

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 .