# 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**C**_{G}(*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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