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Math Help - 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  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  |Orb(s)| \cdot |Stab(s)| = |G| .
    S must be partitioned such that  |Stab(s)| divides  |G| .
    Because p is prime, S must be partitioned into pieces of  a\cdot p^i for some 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  gcd( p , N )\neq 1 or  |Stab(s)| doesn't divide  |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  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  |Orb(s)| \cdot |Stab(s)| = |G| .
    S must be partitioned such that  |Stab(s)| divides  |G| .
    Because p is prime, S must be partitioned into pieces of  a\cdot p^i for some 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  gcd( p , N )\neq 1 or  |Stab(s)| doesn't divide  |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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