Isoclinism and stable cohomology of wreath products

Using the notion of isoclinism introduced by P. Hall for finite p-groups, we show that many important classes of finite p-groups have stable cohomology detected by abelian subgroups, see Theorem 4.4. Moreover, we show that the stable cohomology of the n-fold wreath product of cyclic groups ZZ/p is detected by elementary abelian p-subgroups and we describe the resulting cohomology algebra explicitly. Some applications to the computation of unramified and stable cohomology of finite groups of Lie type are given.