In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators. It is a consequence of the Hall–Witt identity.
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
x
y
.
If H is a subgroup of a group G, then the centralizer of H in G will be denoted by CG(H).
Let X, Y and Z be subgroups of a group G, and assume
[
X
,
Y
,
Z
]
=
1
and
[
Y
,
Z
,
X
]
=
1
Then
[
Z
,
X
,
Y
]
=
1
.
More generally, if
N
◃
G
, then if
[
X
,
Y
,
Z
]
⊆
N
and
[
Y
,
Z
,
X
]
⊆
N
, then
[
Z
,
X
,
Y
]
⊆
N
.
Proof and the Hall–Witt identity
Hall–Witt identity
If
x
,
y
,
z
∈
G
, then
[
x
,
y
−
1
,
z
]
y
⋅
[
y
,
z
−
1
,
x
]
z
⋅
[
z
,
x
−
1
,
y
]
x
=
1.
Proof of the three subgroups lemma
Let
x
∈
X
,
y
∈
Y
, and
z
∈
Z
. Then
[
x
,
y
−
1
,
z
]
=
1
=
[
y
,
z
−
1
,
x
]
, and by the Hall–Witt identity above, it follows that
[
z
,
x
−
1
,
y
]
x
=
1
and so
[
z
,
x
−
1
,
y
]
=
1
. Therefore,
[
z
,
x
−
1
]
⊆
C
G
(
Y
)
for all
z
∈
Z
and
x
∈
X
. Since these elements generate
[
Z
,
X
]
, we conclude that
[
Z
,
X
]
⊆
C
G
(
Y
)
and hence
[
Z
,
X
,
Y
]
=
1
.
