Center of a group

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The center of a mathematical group G is the set, usually denoted Z(G), including each element g of G which commutes with all elements of G. In other words, g is in Z(G) if and only if, for each h in G, gh = hg.

Obviously, Z(G) is a subgroup of G: in fact, if g and g' are in Z(G), then for each h in G

g g' h = g h g' = h g g' ,

so g g' is in Z(G) as well; and a similar argument applies to inverses.

Z(G) is a normal subgroup of G and, moreover, is a characteristic subgroup of it.