Center of a group

The center is always an abelian subgroup of \(G\): it contains the identity, and if \(z,w\) commute with everything, so do \(zw\) and \(z^{-1}\). The two extremes are \(Z(G)=G\), which happens exactly when \(G\) is abelian, and \(Z(G)=\{e\}\), the case of a group with trivial center. Being trivial does not mean the group is «very» nonabelian: only that no element other than the identity commutes with everything.

Examples: \(Z(S_n)=\{e\}\) for \(n\ge 3\). In the dihedral group \(D_n\) of symmetries of the \(n\)-gon, the center is trivial if \(n\) is odd, and has two elements (the identity and the \(180^\circ\) rotation) if \(n\ge4\) is even. In \(GL_n(\mathbb{R})\) the center is the scalar matrices \(\lambda I\) with \(\lambda\ne0\). For a product, the center is the product of the centers.

The center is a very special subgroup: each of its elements commutes with everything, so any subgroup contained in \(Z(G)\) is normal in \(G\). That property makes it useful for forming quotients and detecting structure; for instance, the class equation uses the size of the center to study groups of prime-power order.

A common mistake is to confuse the center with an arbitrary commutative subgroup. A subgroup can be abelian without being central: in \(S_3\), the subgroup generated by a 3-cycle is abelian, but its elements do not commute with the transpositions, so it is not in the center.