Minimalist grammar

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 $[Z,X,Y]=1$.

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

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 $[Z,X,Y]=1$.

See also

• Commutator
• Lower central series
• Grün's lemma

Notes

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