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 .
