What is commutator? — Mathematics

Definition 1.28 University Mathematics — Year 3 · Chapter 1 — Group Theory

The commutator of $x, y \in G$ is $[x,y] = xyx^{-1}y^{-1}$ ; the derived subgroup $D(G)$ is the subgroup generated by all commutators. The derived series is $D^0(G) = G$ , $D^{i+1}(G) = D(D^i(G))$ , and $G$ is solvable if $D^n(G) = \{e\}$ for some $n$ .

Examples

Example 1.30

Abelian groups are solvable. $p$ -groups are solvable, by induction on the order: $Z(G) \neq \{e\}$ and $G/Z(G)$ is a smaller $p$ -group. $S_3$ and $S_4$ are solvable: $S_4 \trianglerighteq A_4 \trianglerighteq V \trianglerighteq \{e\}$ , where $V = \{e, (1\,2)(3\,4), (1\,3)(2\,4), (1\,4)(2\,3)\}$ is the Klein group of double transpositions (normal in $S_4$ : a union of conjugacy classes), with abelian quotients $\Z/2\Z$ , $\Z/3\Z$ , $V$ . In Chapter 4, “the general equation of degree $n$ is solvable by radicals” will literally mean “ $S_n$ is a solvable group”. Whence the importance of the next definition.

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