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Math Help - Group Theory

  1. #1
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    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).)
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  2. #2
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    Quote Originally Posted by Coda202 View Post
    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)).
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  3. #3
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    This is what I meant:

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