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In any group G, some of the elements of G commute with all of the other elements in G

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In any group G, some of the elements of G commute with all of the other elements in G.

The set of all such elements in G is called the centre of G, and is denoted by Z(G).

Hence Z(G) = {g|xg = gx x is a subset of G}.

For instance in any group the identity commutes with every element - so Z(G) is never empty. It should also be clear that:

Z(G) = G if and only if G is Abelian

Prove that this is always true. i.e Prove that in any group G, the centre Z (G) is a subgroup of G. In fact, Z (G) is always a normal subgroup of G (but you are not asked to prove this)