Exercise 7, § 5

Let p be a prime number. (a) Show that $n^p \equiv n \pmod{p}$ for all $n \in \mathbf{Z}$. (Reduce it to the case where n is positive and argue by induction on n using the binomial formula.) (b) Let x, y be two elements of a group G. Suppose that

Show that $y^p xy^{-p} = x^n$; deduce that $y^{p-1}$ commutes with x.

(c) Let G be a group all of whose elements $\neq 1$ are of order p and are conjugate to one another. Show that Card(G) is equal to 1 or 2. (Use (b) to prove that G is commutative.)