1. ## 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).)

2. 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 $\displaystyle G$ be a finite group and let $\displaystyle X$ be the set of all subgroups of $\displaystyle G$. Define an action of $\displaystyle G$ on $\displaystyle X$ by $\displaystyle (g,H)\mapsto gHg^{-1}$ for $\displaystyle g\in G$ and $\displaystyle H\in X$ i.e. action by conjugation. Show that $\displaystyle (~,~): G\times X\to X$ is indeed a group action. Now let $\displaystyle H\in X$ the stabilizer of $\displaystyle H$ is $\displaystyle \{ g\in G | gHg^{-1} = H \}$ i.e. the stabilizer is the normalizer here, so $\displaystyle \text{stab}(H) = N(H)$. Let $\displaystyle H\in X$ then the orbit of $\displaystyle H$ is $\displaystyle \{gHg^{-1} | g\in G \}$ i.e. all the conjugate subgroups to $\displaystyle H$. It follows from the orbit-stabilizer theorem that $\displaystyle \text{orb}(H) = (G:\text{stab}(H)) = (G:N(H))$.

3. This is what I meant:

[IMG]file:///c:/program%20files/literalmath/untitled.html[/IMG]