Commutator subgroup

2020 Mathematics Subject Classification: Primary: 20-XX [MSN][ZBL]

The commutator subgroup of a group $G$, also called the derived group, second term of the lower central series, of $G$, is the subgroup of $G$ generated by all commutators of the elements of $G$ (cf. Commutator). The commutator subgroup of $G$ is usually denoted by $[G,G]$, $G'$ or $\Gamma_2(G)$. The commutator subgroup is a fully-characteristic subgroup, and any subgroup containing the commutator subgroup is a normal subgroup. The quotient group with respect to some normal subgroup is Abelian if and only if this normal subgroup contains the commutator subgroup of the group.

The commutator ideal of a ring $R$ is the ideal generated by all products $ab$, $a,b \in R$; it is also called the square of $R$ and is denoted by $[R,R]$ or $R^2$.

Both the above concepts are special cases of the notion of the commutator subgroup of a multi-operator $\def\O{\Omega}\O$-group $G$, which is defined as the ideal generated by all commutators and all elements of the form

$$\def\o{\omega} -a_1\cdots a_n\o - b_1\cdots b_n\o + (a_1+b_2)\cdots(a_n+b_n)\o, \tag{$*$}$$ where $\o$ is an $n$-ary operation in $\O$ and

$$a_,\dots,a_n,b_1,\dots,b_n\in G.$$

Comments[edit]

In the case of a ring considered as an operator $\O$-group the commutators (of the underlying commutative group) are all zero, so that the commutator ideal is the ideal generated by all elements $-a_1a_2 - b_1b_2 + (a_1+a_2)(b_1+b_2) = a_1b_2 + a_2b_1$. Hence $[R,R]$ is the ideal generated by all products $ab$.

More generally, in all three cases one defines the commutator group (ideal) $[A,B]$ of two $\O$-subgroups $A$ and $B$ as the ideal generated by all commutators $[a,b]$, $a\in A$, $b\in B$, and all elements (*) with $a_1,\dots,a_n\in A$, $b_1,\dots,b_n\in B$.

In the case of a ring $R$ there is a second, different notion which also goes by the name of commutator ideal. It is the ideal generated by all commutators $ab-ba$, $a,b\in R$. This one is universal for homomorphisms of $R$ into commutative rings. I.e. if $\mathfrak{c}$ is this ideal and $\pi : R \to R^{\textrm{ab}} = R/\mathfrak{c}$ is the natural projection, then for each homomorphism $g:R\to A$ into a commutative ring $A$ there is a unique homomorphism $g':R^{\textrm{ab}} \to A$ such that $g = g'\circ \pi$ ($g$ factors uniquely through $\pi$). This is analogous to the property that for ordinary groups $G\to G^{\textrm{ab}} = G/[G,G]$ is universal for mappings of $G$ into Abelian groups (cf. Universal problems).

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