8 - Finite groups with p-rank ≤ 2

Summary

At the end of Chapter 6 we gave a number of examples to show that it is possible for the subring of Chern classes to exhaust the even dimensional cohomology of a finite group. We propose to examine this phenomenon more carefully; as the examples suggest the relative ‘size’ of Ch(G) is closely connected to the abelian subgroup structure of G.

Definition

The p-rank (rk_p) of the finite group G equals the dimension over F_p of a maximal elementary abelian p-subgroup K of G. Thus if K ≅ (ℤ/p) × … × (ℤ/p) (r copies), rk_p(G) = r.

If rk_p(G) = 1, a p-Sylow subgroup is either cyclic or generalised quaternion/binary dihedral (if p = 2). Thus rk_p(G) = 1 if and only if H^*(G, ℤ)_(p) is periodic, see the final section of Chapter 3.

Abelian groups

Theorem 8.1

1. (i) H^*(C_p, ℤ) = (ℤ/p) [α], dim (α) = 2.

2. (ii) H^*(C_p × C_p, ℤ) = (ℤ/p) [α, β] ⊗ E(µ) if p is an odd prime, dim (α) = dim (β) = 2, and dim (µ) = 3.

3. (iii) H^even(C_p × C_p × C_p, ℤ) is generated over ℤ/p by elements α, β and γ of dimension 2 and by ξ of dimension 4.

Proof. Part (i) is familiar; parts (ii) and (iii) follow from the diagram

in which the two monomorphisms are reduction mod p.