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

3. This is what I meant:

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