Kommutaatiojoukon

A Kommutaatiojoukon is the Finnish term for the commutator subgroup of a group, also known in English as the derived subgroup. It is a fundamental construct in group theory that captures the essence of a group's non‑abelian structure. For a group \(G\), the commutator subgroup, denoted \([G,G]\) or \(G'\), is generated by all commutators \([a,b]=aba^{-1}b^{-1}\) where \(a,b\) are elements of \(G\). Every element of \([G,G]\) can be expressed as a finite product of such commutators and their inverses.

The commutator subgroup is a normal subgroup of \(G\), and the quotient \(G/[G,G]\) is always abelian. This fact provides a canonical way to measure how far a group is from being abelian: the larger the commutator subgroup, the more non‑commutative the group. In particular, \(G\) is abelian precisely when \([G,G]\) is trivial, i.e., equals the identity element.

This subgroup plays a key role in the construction of derived series, a descending chain of subgroups used to define solvable groups. A group is solvable if its derived series terminates in the trivial subgroup after finitely many steps. Consequently, the commutator subgroup is often called the first derived subgroup.

Examples include the symmetric group \(S_n\) for \(n\geq3\), whose commutator subgroup is the alternating group \(A_n\). For the quaternion group \(Q_8\), the commutator subgroup is the subgroup of order two generated by \(-1\), reflecting the group's nilpotency. In finite groups, the index \([G:[G,G]]\) equals the size of the largest abelian quotient, linking the theory to group actions and representation theory.