Three subgroups lemma

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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.[1]

More generally, if N\triangleleft G, then if [X,Y,Z]\subseteq N and [Y,Z,X]\subseteq N, then [Z,X,Y]\subseteq N.[2]

Proof and the Hall–Witt identity[edit]

Hall–Witt identity

If x,y,z\in G, then

[x, y^{-1}, z]^y\cdot[y, z^{-1}, x]^z\cdot[z, x^{-1}, y]^x = 1

Proof of the Three subgroups lemma

Let x\in X, y\in Y, and z\in 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}]\subseteq \bold{C}_G(Y) for all z\in Z and x\in X. Since these elements generate [Z,X], we conclude that [Z,X]\subseteq \bold{C}_G(Y) and hence [Z,X,Y]=1.

See also[edit]


  1. ^ Isaacs, Lemma 8.27, p. 111
  2. ^ Isaacs, Corollary 8.28, p. 111