Encyclopedia > Center of a group

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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.

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