How do you show that the centre of a group is a subgroup

To show something is a subgroup we need to show that it satisfies the group axioms. Therefore we need to show that if g and h are in Z(G) then gh is in Z(G), g^-1 is in Z(G), the identity e is in Z(G). As eg = g = ge for all elements g in G we can see e is in Z(G). Then suppose we have g and h in Z(G). Then for all elements j in G we have ghj = gjh as h is in Z(G) = jgh as g is in Z(G). Therefore Z(G) is closed under the group operation. Also we have g^-1 j = g^-1 j e as e is the identity = g^-1 j g g^-1 by definition of inverses = g^-1 g j g^-1 as g is in Z(G) = e j g^-1 = j g^-1 and so g^-1 is in Z(G) and so Z(G) is closed under inverses and is therefore a subgroup of G