Counterexamples to a fiber theorem

Elementary abelian subgroups in p-groups with a cyclic derived subgroup

“…By Lemma 4.3, we have that P contains a normal elementary abelian subgroup E 0 of rank 2 containing Z . Let M = C P (E 0 ), Proposition 4.1 gives an isomorphism The preceding proposition tells us that the dimension of the spheres occurring in A p (P ) 2 in its homotopy type.…”

“…In [6], Fumagalli claimed that Question 1.1 has a positive answer if the group G is solvable. Unfortunately, his proof relies on a result that turns out to be false (see [2]), so that to our knowledge the following question seems to remain opened. In [11], Pulkus and Welker showed that for solvable groups G, the study of A p (G) can be reduced to the study of upper intervals A p (C G (A)) >A .…”

“…In Section 4, we review the results of Bouc and Thévenaz and show how they can be used effectively in some cases to determine the homotopy type of A p (P ) 2 . More precisely, we compute the number of spheres occurring in each dimension in the homotopy type of A p (P ) 2 when P is a p-group with a cyclic Frattini subgroup.…”

“…More precisely, we compute the number of spheres occurring in each dimension in the homotopy type of A p (P ) 2 when P is a p-group with a cyclic Frattini subgroup. As a consequence, we obtain that Question 1.3 has a positive answer if P has a cyclic derived subgroup.…”

“…In Section 5, we give first a sharp upper bound on the dimension of the spheres occurring in the homotopy type of A p (P ) 2 . We consider then the problem of describing the p-groups for which this bound is reached.…”