Let X, Y and Z be subgroups of a group G, and assume
- and
Then .[1]
More generally, for a normal subgroup of , if and , then .[2]
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 .