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Let G be a non-abelian group of order p^3 with p prime. (a) Show that |Z(G)|...

Let G be a non-abelian group of order p^3 with p prime.

(a) Show that |Z(G)| = p. (b) Suppose a /∈ Z(G). Show that |NG(a)| = p^2 .

(c) Show that G has exactly p 2 +p−1 conjugacy classes (don’t forget to count the classes of the elements of Z(G)).

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