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Let G be a group. Define Z(G) ={x∈G|xg=gx for all g∈G}, that is Z(G) is the...

Let G be a group. Define Z(G) ={x∈G|xg=gx for all g∈G}, that is Z(G) is the set of elements commuting with all the elements of G. We call Z(G) the center of G. (In German, the word for

center is Zentrum, hence the use of the “Z”.)

(a) Show that Z(G) is a subgroup of G.

(b) Show that Z(G) is an abelian group.

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