The first purely algebraic proof of this fact seems to be from Leonard Evens:
- A Generalization of the Transfer Map in the Cohomology of Groups, Trans. Amer. Math. Soc. 108(1963), 54-65 [Theorem 3]
where he proves the result with help of his norm map. After having established the basic properties of the norm map, the proof is rather elementary. : Let $C$ be a cyclic subgroup of prime order of $G$ and let $x$ be a generator of $H^2(C,\mathbb{Z})$. Then the powers of $y = N^G_C(x)$ yield non-trivial cohomology classes of $H^*(G,\mathbb{Z})$ in degrees divisible by $(G:C)$.
From a historical point of view the norm map already occured in disguise in Evens´ paper
- The Cohomology Ring of a Finite Group, Trans. Amer. Math. Soc. 101(1961), 224-239
where he proves finte generation of the cohomology ring.
