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Thread: Group action on set and fixed point

  1. #1
    Senior Member vincisonfire's Avatar
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    Group action on set and fixed point

    Question : Let p be a prime number. Let G be a finite group of $\displaystyle p^r $ elements. Let S be a finite set having N elements and assume that gcd( p , N ) = 1. Assume that G acts of S. Prove that G has a fixed point in S.
    Answer : It is possible to prove that $\displaystyle |Orb(s)| \cdot |Stab(s)| = |G| $.
    S must be partitioned such that $\displaystyle |Stab(s)| $ divides $\displaystyle |G| $.
    Because p is prime, S must be partitioned into pieces of $\displaystyle a\cdot p^i $ for some $\displaystyle a,i \in \mathbb N^* $
    But we know that gcd( p , N ) = 1. Therefore, G must have at least one fixed element. Else, we get a contradiction that is $\displaystyle gcd( p , N )\neq 1 $ or $\displaystyle |Stab(s)| $ doesn't divide $\displaystyle |G| $.
    I'm wondering if my argument is sufficient.
    I like math, but I don't have the spirit of a mathematician so I'm a little insecure. Thanks for your time.
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  2. #2
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    Quote Originally Posted by vincisonfire View Post
    Question : Let p be a prime number. Let G be a finite group of $\displaystyle p^r $ elements. Let S be a finite set having N elements and assume that gcd( p , N ) = 1. Assume that G acts of S. Prove that G has a fixed point in S.
    Answer : It is possible to prove that $\displaystyle |Orb(s)| \cdot |Stab(s)| = |G| $.
    S must be partitioned such that $\displaystyle |Stab(s)| $ divides $\displaystyle |G| $.
    Because p is prime, S must be partitioned into pieces of $\displaystyle a\cdot p^i $ for some $\displaystyle a,i \in \mathbb N^* $
    But we know that gcd( p , N ) = 1. Therefore, G must have at least one fixed element. Else, we get a contradiction that is $\displaystyle gcd( p , N )\neq 1 $ or $\displaystyle |Stab(s)| $ doesn't divide $\displaystyle |G| $.
    I'm wondering if my argument is sufficient.
    I like math, but I don't have the spirit of a mathematician so I'm a little insecure. Thanks for your time.
    A solution can be found in a very old post here.
    (It seems you are using the same book).
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  3. #3
    Senior Member vincisonfire's Avatar
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    It's course notes from Eyal Goren. Very nice of him to give them to us updated each week. Maybe you know him by name. He's a very nice teacher.
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