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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