# Thread: Group action on set and fixed point

1. ## 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.

2. Originally Posted by vincisonfire
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).

3. 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.