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**Coda202** Let G be a finite group with g in G. Let N(G) be the centralizer of g in G be defined as {x|xgx^-1 = g}

Prove that N(G) is a subgroup of G

Letting S(G) = {xgx^-1|x in G}

Prove that |S(G)| = |G/N(G)|

and prove that {S(G)|g in G} is a partition of G

Define Z(G) to be the family of intersecting subgroups (g in G) of N(G).

Prove that if g in Z(G) then |S(G)| = 1

Finally, show that:

|G| = |Z(G)| + the sum of |S(G)| using (g in G) - Z(G) (where the sum is over the sizes of distinct S(G).)