Editing Group ring (section)

===Center of a group algebra===
The [[center of a group|center]] of the group algebra is the set of elements that commute with all elements of the group algebra:
:<math>\mathrm{Z}(K[G]) := \left\{ z \in K[G] : \forall r \in K[G], zr = rz \right\}.</math>

The center is equal to the set of [[class function]]s, that is the set of elements that are constant on each conjugacy class
:<math>\mathrm{Z}(K[G]) = \left\{ \sum_{g \in G} a_g g :  \forall g,h \in G, a_g = a_{h^{-1}gh}\right\}.</math>

If {{nowrap|1=''K'' = '''C'''}}, the set of irreducible [[character theory|characters]] of ''G'' forms an orthonormal basis of Z(''K''[''G'']) with respect to the inner product
:<math>\left \langle \sum_{g \in G} a_g g, \sum_{g \in G} b_g g \right \rangle = \frac{1}{|G|} \sum_{g \in G} \bar{a}_g b_g.</math>