Three subgroups lemma

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 [H, K], 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 [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).

Let X, Y and Z be subgroups of a group G, and assume

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

Then $[Z,X,Y]=1$.Isaacs, Lemma 8.27, p. 111

More generally, for a normal subgroup $N$ of $G$, if $[X,Y,Z]\backslash subseteq\; N$ and $[Y,Z,X]\backslash subseteq\; N$, then $[Z,X,Y]\backslash subseteq\; N$.Isaacs, Corollary 8.28, p. 111

Hall–Witt identity

If $x,y,z\backslash in\; G$, then

: $[x,\; y^\{-1\},\; z]^y\backslash cdot[y,\; z^\{-1\},\; x]^z\backslash cdot[z,\; x^\{-1\},\; y]^x\; =\; 1.$

Proof of the three subgroups lemma

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

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

{{Reflist|2}}

• {{cite book

| author = I. Martin Isaacs

| author-link = Martin Isaacs

| year = 1993

| title = Algebra, a graduate course

| edition = 1st

| publisher = Brooks/Cole Publishing Company

| isbn = 0-534-19002-2

}}

Category:Lemmas in group theory

Category:Articles containing proofs