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**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.