Suppose that G is Abelian group of order 35 and every element of G satisfies the equation x^35=e. Prove that G is cyclic
The condition $\displaystyle x^{35} = e$ is absolutely unnecessary because that is just Lagrange's theorem.
Since $\displaystyle 35 = 5\cdot 7$ for two distinct primes it means that is must be cyclic.