Thread: Group Theory

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).)

Originally Posted by 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).)

I think you are confused here, also you notation is not good.

Let be a finite group and let be the set of all subgroups of . Define an action of on by for and i.e. action by conjugation. Show that is indeed a group action. Now let the stabilizer of is i.e. the stabilizer is the normalizer here, so . Let then the orbit of is i.e. all the conjugate subgroups to . It follows from the orbit-stabilizer theorem that .

This is what I meant:

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