Volume 8, Issue 6

Let G   be a finite p  -group. We show that if Ω 2 ( G  ) is an extraspecial group then Ω 2 ( G  ) =  G  . If we assume only that (the subgroup generated by elements of order p   2  ) is an extraspecial group, then the situation is more complicated. If p  = 2, then either  =  G   or G   is a semidihedral group of order 16. If p  > 2, then we can only show that  =  H p ( G  ).

We give upper and lower bounds for the number of isomorphism types of pro- p  -groups of coclass r , or, equivalently, for the number of infinite branches in the graph of all finite p  -groups of coclass r . As a step towards this goal, we determine bounds for the number of isomorphism types of uniserial p  -adic space groups of coclass r .

Let  = ( q  ) be the group of all upper unitriangular matrices of size n over , the field with q  =  p t elements. We show that the computation of many canonical representatives of the conjugacy classes of can be reduced to similar calculations for certain matrices of smaller size that we call condensed matrices. Our methods make use of combinatorial techniques including (polynomial) generating functions, and they greatly increase the efficiency of the calculations of the conjugacy vector of . They also allow us to obtain the number of conjugacy classes of size q  z for any z  ≤ 2 n  − 8. These numbers are polynomial functions of q , in accordance with a well-known conjecture of G. Higman.

We study the connectivity of the coset poset and the subgroup poset of a group, focusing in particular on simple connectivity. The coset poset was recently introduced by K. S. Brown in connection with the probabilistic zeta function of a group. We take Brown’s study of the homotopy type of the coset poset further, and in particular generalize his results on direct products and classify direct products with simply connected coset posets. The homotopy type of the subgroup poset L ( G  ) has been examined previously by Kratzer, Thévenaz, and Shareshian. We generalize some results of Kratzer and Thévenaz, and determine π 1 ( L ( G  )) in nearly all cases.

It is shown that, for any pair of cardinals with infinite sum, there exist a group and an equation over this group such that the first cardinal is the number of solutions to this equation and the second cardinal is the number of non-solutions. A countable torsion-free nontopologizable group is constructed.

Using methods of combinatorial group theory, we give an elementary construction of polyhedra whose links are (not necessarily isomorphic) connected bipartite graphs. In particular, we construct polyhedra whose links are generalized m  -gons. Polyhedra of this type are interesting because their universal coverings are two-dimensional hyperbolic buildings with different links. We show that the fundamental groups of some of our polyhedra contain surface groups. The presentation of the results is given in the language of combinatorial group theory.

We introduce generic units in ℤ C n and prove that they are precisely the shifted cyclotomic polynomials. They generate the group  of constructible units. For each cyclic group we produce a basis of a finite index subgroup of integral units consisting of certain irreducible cyclotomic polynomials; this extends a result of Hoechsmann and Ritter. We also study ‘alternating-like’ units and decide when they generate a subgroup of finite index.