Three subgroups lemma

In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators. It is a consequence of the Hall–Witt identity.

Notation

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 C_G(H).

Statement

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

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, $[z,x^{{-1}}]\subseteq {\mathbf {C}}_{G}(Y)$ for all $z\in Z$ and $x\in X$ . Since these elements generate $[Z,X]$ , we conclude that $[Z,X]\subseteq {\mathbf {C}}_{G}(Y)$ and hence $[Z,X,Y]=1$ .

Notes

↑ Isaacs, Lemma 8.27, p. 111
↑ Isaacs, Corollary 8.28, p. 111

References

I. Martin Isaacs (1993). Algebra, a graduate course (1st ed.). Brooks/Cole Publishing Company. ISBN 0-534-19002-2.

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