Constructing modular classifying spaces

Chevalleyp–local finite groups

“…Part (2) of Theorem E also explains the often observed fact that finite Chevalley groups G.q/ and G.q 0 / have same cohomology ring or identical p -local structure when q and q 0 are prime powers, with q r Á q 0 r Á 1 mod p and p .1 q r / D p .1 q 0 r /, for some r , 1 Ä r Ä p 1. We plan to investigate this phenomenon closer in a future paper.…”

“…Later, Aguadé gave a nice uniform construction of a family of modular p -compact groups including these cases [2].…”

“…In cases (1) and (3) we always obtain that BX.q/ is the p -completed classifying space of a finite group. The other two families contain the new exotic examples of p -local finite groups.…”

“…It turns out that X 12 and X 31 are the two exotic p -compact groups originally constructed by Zabrodsky [71], and later included in the Aguadé family [2]. Zabrodsky used the actions of these same automorphisms, ' and i , on the homotopy groups of BF 4 and BE 8 , respectively, and realized the invariant subgroups as homotopy groups of new spaces, BX 12 and BX 31 .…”

“…Since we can decompose a p -adic unit q as q D q 0 where is a .p 1/-st root of unity and q 0 Á 1 mod p , part (1) of Theorem E will reduce the question of computing BX.q/ to the case where q Á 1 mod p which turns out to be easier to handle in abstract calculations and concrete examples. The second part of the theorem tells us that BX.q/ does only depend on the order r of q mod p and the p -adic valuation p .1 q r /, so we can change the exact value of q at our convenience if we keep those parameters fixed.…”