Variations on average character degrees and p-nilpotence

CHARACTER DEGREES AND NILPOTENCE CLASS OF FINITE p-GROUPS: AN APPROACH VIA PRO-p GROUPS

Let S be a finite set of powers of p containing 1. It is known that for some choices of S, if P is a finite p-group whose set of character degrees is S, then the nilpotence class of P is bounded by some integer that depends on S, while for some other choices of S such an integer does not exist. The sets of the first type are called class bounding sets. The problem of determining the class bound...

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...

On Finite Groups With Two Irreducible Character Degrees

AFFINE SUBGROUPS OF THE CLASSICAL GROUPS AND THEIR CHARACTER DEGREES

In this paper we describe how the degrees of the irreducible characters of the affine subgroups of the classical groups under consideration can be found inductively. In [4] Gow obtained certain character degrees for all of the affine subgroups of the classical groups. We apply the method of Fischer to the above groups and, in addition to the character degrees given in [4], we obtain some ne...

The Nilpotence Height of P

The method of Walker and Wood is used to completely determine the nilpotence height of the elements P s t in the Steenrod algebra at the prime 2. In particular, it is shown that (P s t ) 2bs/tc+2 = 0 for all s ≥ 0, t ≥ 1. In addition, several interesting relations in A are developed in order to carry out the proof.