The Quillen conjecture asserts that contractibility of the order complex of the nonidentity elementary
Abelian p-subgroups of a finite
group 
implies a nontrivial normal p-subgroup. Here 
is prime,
is ordered by inclusion, and its 0-dimensional simplices
correspond to subgroups isomorphic
to 
with 
.
Writing 
for the largest normal p-subgroup, the implication
is
 |
The converse holds as well.
