Encyclopedia > Center of a group

  Article Content

Center of a group

In abstract algebra, the center (or centre) of a group G is the set, usually denoted Z(G), which includes each element z of G which commutes with all elements of G. In other words, z is in Z(G) if and only if, for each g in G, gz = zg.

Z(G) is a subgroup of G: in fact, if x and y are in Z(G), then for each g in G

(xy)g = x(yg) = x(gy) = (xg)y = (gx)y = g(xy)
so xy is in Z(G) as well; and a similar argument applies to inverses.

Moreover, Z(G) is an abelian subgroup of G, and a normal subgroup of G, and even a characteristic subgroup of it.

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
Westhampton Beach, New York

... out of which 22.2% have children under the age of 18 living with them, 47.1% are married couples living together, 11.1% have a female householder with no husband ...

This page was created in 38.4 ms