The main aim of this paper is to find necessary and sufficient conditions under which a square matrix over a ring is a semicentral idempotent matrix. Using & ell; -braided and r -braided idempotents of the ring, we construct new classes of idempotent matrices that are not semicentral idempotents.
Let p be an arbitrary prime number and let P be a finite p -group. Let Ap(P)>= 2 be the poset of all elementary abelian subgroups of P of rank at least 2. Bouc and Th & eacute;venaz proved that Ap(P)>= 2 has the homotopy type of a wedge of spheres (of possibly different dimensions). In this paper, we firstly classify finite 2-groups P with both cyclic center and P 'congruent to C2 & times;C2 , then compute exactly the homotopy type of Ap(P)>= 2 and prove that Ap(P)>= 2 is contractible in this case.
