Basic p-groups

Basic Theory of Algebraic Groups

Some Properties of p-Groups and Commutative p-Groups

For simplicity, we use the following convention: G is a group, a, b are elements of G, m, n are natural numbers, and p is a prime natural number. One can prove the following propositions: (1) If for every natural number r holds n 6= pr, then there exists an element s of N such that s is prime and s | n and s 6= p. (2) For all natural numbers n, m such that n | pm there exists a natural number r...

Character degrees of p-groups and pro-p groups

In the 1970s, Isaacs conjectured that there should be a logarithmic bound for the length of solvability of a p-group G with respect to the number of different irreducible character degrees of G. So far, there are just a few partial results for this conjecture. In this note, we say that a pro-p group G has property (I) if there is a real number D = D(G) that just depends on G such that for any o...

Finite Groups with p-Sylow Coverings

Limit Groups for Relatively Hyperbolic Groups, I: the Basic Tools

We begin the investigation of Γ-limit groups, where Γ is a torsion-free group which is hyperbolic relative to a collection of free abelian subgroups. Using the results of [16], we adapt the results from [22]. Specifically, given a finitely generated group G, and a sequence of pairwise non-conjugate homomorphisms {hn : G → Γ}, we extract an R-tree with a nontrivial isometric G-action. We then pr...