Let G be a group and let o G times G rightarrow G a b righta

Let G be a group and let o :G times G rightarrow G, (a, b) rightarrow aba^-1 be the action of G on G by conjugation. Define Z(G) := {a elementof G | ab = ba for all b elementof G}. (a) Show that Z(G) = Fix_G^Phi(G). (b) Suppose G is a p-group with |G| > 1. Show that |Z(G)| > 1.

Solution

(a) By definition,

           Z(G) = { a in G| ab = ba for all b in G}

Now Fix _C (G) = { a in G| gag^-1 = a for all g in G}

                      = { a in G| ga =ag for all g in G}

                      =Z(G)

So (a) is proved.

(b) Fixed point formula (or the class equation) asserts

|G| = |Z(G)| + |G:C_G (x)|

x running through representatives of distinct conjugacy classes (away from the center Z)

If G is a p-group, LHS is divisible by p and every term on the second sum on the RHS is divisible by p, as

each is the index of a (non trivial) subgroup of G .

Thus forces p to divide |Z(G)\\. But |Z(G)| is at least 1. this implies |Z(G)| >1 . Hence the result.