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In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators. It is a consequence of Philip Hall and Ernst Witt's eponymous identity.

Notation

In what follows, the following notation will be employed:

• If H and K are subgroups of a group G, the commutator of H and K, denoted by , is defined as the subgroup of G generated by commutators between elements in the two subgroups. If L is a third subgroup, the convention that = ,L] will be followed.
• If x and y are elements of a group G, the conjugate of x by y will be denoted by ${\displaystyle x^{y}}$.
• 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

${\displaystyle =1}$ and ${\displaystyle =1.}$

Then ${\displaystyle =1}$.

More generally, for a normal subgroup ${\displaystyle N}$ of ${\displaystyle G}$, if ${\displaystyle \subseteq N}$ and ${\displaystyle \subseteq N}$, then ${\displaystyle \subseteq N}$.

Proof and the Hall–Witt identity

Hall–Witt identity

If ${\displaystyle x,y,z\in G}$, then

${\displaystyle ^{y}\cdot ^{z}\cdot ^{x}=1.}$

Proof of the three subgroups lemma

Let ${\displaystyle x\in X}$, ${\displaystyle y\in Y}$, and ${\displaystyle z\in Z}$. Then ${\displaystyle =1=}$, and by the Hall–Witt identity above, it follows that ${\displaystyle ^{x}=1}$ and so ${\displaystyle =1}$. Therefore, ${\displaystyle \in \mathbf {C} _{G}(Y)}$ for all ${\displaystyle z\in Z}$ and ${\displaystyle x\in X}$. Since these elements generate ${\displaystyle }$, we conclude that ${\displaystyle \subseteq \mathbf {C} _{G}(Y)}$ and hence ${\displaystyle =1}$.

See also

• Commutator
• Lower central series
• Grün's lemma
• Jacobi identity

Notes

1. Isaacs, Lemma 8.27, p. 111
2. 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.

• Lemmas in group theory