In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators. It is a consequence of Philip Hall and Ernst Witt's eponymous identity.
In what follows, the following notation will be employed:
Let X, Y and Z be subgroups of a group G, and assume
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 .
In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators. It is a consequence of Philip Hall and Ernst Witt's eponymous identity.
In what follows, the following notation will be employed:
Let X, Y and Z be subgroups of a group G, and assume
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 .