Let P be a fixed p-group for an odd prime. We are interested in the localized
classifying spaces BG(p) (hereafter denoted simply by BG) for various groups G
sharing the same Sylow p-subgroup P. If such groups G become bigger then BG
becomes smaller, because a transfer argument shows that BG is a stable summand of
BP and of intermediate subgroups. For this reason, C. B. Thomas [16] first found
that, in many cases, the odd component of the cohomology of sporadic simple groups
should be simple even if the cohomology of P is quite complicated. In particular,
when G is the biggest Janko group J4, D. J. Green [6]
showed that Heven(G)/3 is the
Dickson algebra of rank two. The Janko group has a Sylow 3-subgroup
31+2+ = E, the extra special 3-group of order 33
and of exponent 3. Tezuka and Yagita [14] studied
the even-dimensional cohomology of all sporadic simple groups with
[mid ]P[mid ] = p3; indeed,
in these cases P is isomorphic to the extra-special group
p1+2+ = E.
In this paper, we study BG for simple groups G with a Sylow p-subgroup E. We
note the importance of G-conjugacy classes of p-pure elementary abelian p-subgroups
of rank two. Here ‘p-pure’ means that all nonzero elements in the subgroup are G-conjugate. In Section 1, we see that BG is expressed as a homotopy pushout of
B(NG(E)) and of B(p-pure
subgroup), when NG(E)
= NG(Z(E)). The last condition is
always satisfied if p>3. In Section 2, the case p = 3 is studied; for example, we see
the homotopy equivalence BJ4 ≅ BRu. In Section 3, we show that stable homotopy
splitting is given from the dominant summands of E and of non-p-pure subgroups.
When p = 3 the splitting is given explicitly; for example, BJ4 is the summand induced
from the trivial representation in F3[Out(E)]. In the last section, we add the list of
sporadic simple groups (due to Yoshiara) and cohomology
H*(G)/√0 for p = 3 from
the paper [14] for the reader's convenience. The author thanks Satoshi Yoshiara, who
pointed out to him the paper of Benson and taught him the importance of p-pure
subgroups. He also thanks Michishige Tezuka; indeed, most of the results in this
paper are natural consequences of results in the joint paper [14] with Tezuka.