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
North Haven, New York

... age 18 and over, there are 82.7 males. The median income for a household in the village is $74,583, and the median income for a family is $81,363. Males have a media ...

 
 
 
This page was created in 24.6 ms