TITLE: Non-vanishing of group cohomology in sufficiently high degree
QUESTION [21 upvotes]: Atiyah in his famous paper , Characters and cohomology of finite groups, after proving completion of representation ring in augmentation ideal is the same as $ K(BG)$, gives bunch of corollaries of this main theorem. One of them that catches my interest is: For any finite non-trivial group $G$ there exists arbitrary large integer $n$ such that $H^n(G,\mathbb{Z})\neq 0 $. I just wonder if anyone can prove this without this powerful theorem.

REPLY [11 votes]: 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.