A commutator measures the failure of group elements to commute. Commutator subgroups, derived series, and lower central series control abelianization, solvability, and nilpotence respectively. Commutator identities are also basic computational tools in finite \(p\)-groups, coprime actions, transfer, and local group theory. [@isaacs2008]

Definition and convention

Some sources use the inverse convention \(xyx^{-1}y^{-1}\). One must check conventions before transferring formulas, because inverses and conjugating positions change.

For subgroups \(H,K\leq G\), define

\[ [H,K]=\langle[h,k]:h\in H,\ k\in K\rangle. \]

This is the subgroup generated by the commutators, not generally the set of commutators itself. In particular, elements of \(G'=[G,G]\) need not each be a single commutator.

Basic identities

With the convention above,

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

and \([x^g,y^g]=[x,y]^g\).

Quotients, normalization, and centralization

If \(N\triangleleft G\), then

\[ [HN/N,KN/N]=[H,K]N/N. \]

Thus the images of \(H\) and \(K\) centralize one another in \(G/N\) precisely when \([H,K]\leq N\). Also,

\[ K\leq N_G(H)\quad\Longleftrightarrow\quad[H,K]\leq H. \]

In particular, \(H\triangleleft G\) exactly when \([H,G]\leq H\). If \(H\) and \(K\) normalize one another and intersect trivially, then \([H,K]=1\) and they centralize one another elementwise.

Derived subgroup and abelianization

The quotient \(G/G'\) is the abelianization. Every homomorphism from \(G\) to an abelian group factors uniquely through it.

The derived series is

\[ G^{(0)}=G,\qquad G^{(n+1)}=[G^{(n)},G^{(n)}]. \]

\(G\) is solvable precisely when \(G^{(n)}=1\) for some \(n\).

Multiple commutators and the lower central series

Use left association:

\[ [x_1,x_2,\ldots,x_n]=[[x_1,x_2,\ldots,x_{n-1}],x_n]. \]

Define

\[ \gamma_1(G)=G,\qquad\gamma_{n+1}(G)=[\gamma_n(G),G]. \]

Isaacs writes \(G^n\) for \(\gamma_n(G)\); the gamma notation avoids confusion with power subgroups.

It follows that \(G^{(r)}\leq\gamma_{2^r}(G)\). If \(G\) has nilpotency class \(c\), its derived length is at most \(1+\lfloor\log_2c\rfloor\).

Three-subgroup lemma

This subgroup form of the Hall–Witt identity is used to prove lower-central estimates and appears throughout coprime action and local group theory.

Groups of class two

If \(G\) has nilpotency class at most two, then \(G'\leq Z(G)\), so all commutators are central. Consequently

\[ [xy,z]=[x,z][y,z] \]

and

\[ (xy)^n=x^ny^n[y,x]^{\binom n2}. \]

For an odd prime \(p\), if a class-two \(p\)-group has \(P'\) of exponent dividing \(p^e\), then \(P/Z(P)\) also has exponent dividing \(p^e\), because

\[ [x^{p^e},y]=[x,y]^{p^e}=1. \]

Abelian normal subgroup with cyclic quotient

Collection process

Commutator collection rewrites group words as ordered generator powers followed by commutators of increasing weight. In nilpotent groups sufficiently high-weight commutators vanish, so the process terminates. It is used to:

  • derive correction terms in \((xy)^n\);
  • control exponents and nilpotency classes of \(p\)-groups;
  • compare derived and central series;
  • prove normal-complement and coprime-action theorems;
  • construct polynomial coordinates for nilpotent groups.

Common errors

The general algebraic scope was cross-checked against Wikipedia's “Commutator” article. Group-theoretic conventions, results, and proofs were organized from Isaacs, Chapter 4. The cited revision is attributed under CC BY-SA. [@wikipedia-commutator]