(It seems you are using the same book).
Question : Let p be a prime number. Let G be a ﬁnite group of 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 .
S must be partitioned such that divides .
Because p is prime, S must be partitioned into pieces of for some
But we know that gcd( p , N ) = 1. Therefore, G must have at least one fixed element. Else, we get a contradiction that is or doesn't divide .
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.