Groups with two extreme character degrees and their normal subgroups

Abstract

Abstract. We study the finite groups G for which the set cd(G) of irreducible complex character degrees consists of the two most extreme possible values, that is, 1 and |G : Z(G)| 1/2 . We are easily reduced to finite p-groups, for which we derive the following group theoretical characterization: they are the p-groups such that |G : Z(G)| is a square and whose only normal subgroups are those containing G or contained in Z(G). By analogy, we also deal with pgroups such that |G : Z(G)| = p 2n+1 is not a square, and we prove that cd(G) = {1, p n } if and only if a similar property holds: for any N G,The proof of these results requires a detailed analysis of the structure of the p-groups with any of the conditions above on normal subgroups, which is interesting for its own sake. It is especially remarkable that these groups have small nilpotency class and that, if the nilpotency class is greater than 2, then the index of the centre is small, and in some cases we may even bound the order of G.

Characters of p-groups and Sylow p-subgroups

“…With this notation, the groups studied in [4] are exactly those that satisfy m 1 (P ) = 0. One of the main results of that paper characterizes such groups in terms of their normal subgroups.…”

“…Now we prove that if P is a group of class 2 and a(P ) is small, then m 1 (P ) is small. We first need a lemma, which is a generalization of [4,Theorem D]. In order to prove it it is enough to mimic the proof of Theorem D of [4], so we will just give a sketch of it.…”

“…We recall from [4] the definition of the strong (weak) condition. We say that a p-group P satisfies the strong (weak) condition on normal subgroups if for every N P , either P ≤ N or N ≤ Z(P ) (either P ≤ N or |N Z(P ) :…”

“…This analysis leads G. A. Fernández-Alcober and the author to grouptheoretical properties of the groups under consideration which, we think, have some interest by themselves. For instance, in Theorem F of [4] we obtain a bound for the index of the center of the groups with any of these properties and class ≥ 3. These definitions and part of Theorem F of [4] have been generalized by Isaacs [14].…”

“…For instance, in Theorem F of [4] we obtain a bound for the index of the center of the groups with any of these properties and class ≥ 3. These definitions and part of Theorem F of [4] have been generalized by Isaacs [14]. Isaacs defines an invariant a(P ) associated to any finite p-group P as follows: a(P ) = a is the minimum integer such that if N is a normal subgroup of P and |N Z(P ) : Z(P )| ≥ p a then P ≤ N .…”