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
{\displaystyle [Z,X,Y]=1}
.
[1] More generally, for a normal subgroup
{\displaystyle N}
of
, if
[
,
[2] Hall–Witt identity If
, then Proof of the three subgroups lemma Let
, and by the Hall–Witt identity above, it follows that
Since these elements generate
, we conclude that