Thomas R. Wolf

Non-divisibility among character degrees

Abstract In this paper, we study groups for which if 1 < a < b are character degrees, then a does not divide b. We say that these groups have the condition no divisibility among degrees (NDAD). We conjecture that the number of character degrees of a group that satisfies NDAD is bounded and we prove this for solvable groups. More precisely, we prove that solvable groups with NDAD have at most four character degrees and have derived length at most 3. We give a group-theoretic characterization of the solvable groups satisfying NDAD with four character degrees. Since the structure of groups with at most three character degrees is known, these results describe the structure of solvable groups with NDAD.

Brauer characters of q′-degree in p-solvable groups

Let G be a finite group and let p be a prime. We let Irr(G) and IBr,(G) denote the ordinary irreducible characters and the irreducible Brauer characters of G. Then p 1 x( 1) for all x E Irr(G) if and only if G has normal abelian Sylow-p-subgroup. For p-solvable G, this is a well-known result of Ito. Michler [9] recently proved the general case using the classification of simple groups. Also G has a normal Sylow-p-subgroup if and only if p j j?( 1) for all fl E IBr(G). This is due to Okuyama [ 1 l] for p = 2 and Michler [lo] for p odd. Suppose that G is p-solvable, q is a prime different from p, and q I/?( 1) for all j3 E IBrJG). We show that O”‘(G) is indeed solvable and that the q-length of G is at most two. In particular, since q-factors of G are necessarily abelian, it follows that the Sylow-q-subgroups of G are metabelian. For groups of odd order, these bounds were first obtained by Manz [S]. We also show that the p-length of P’(G) is bounded by at most

Character degrees and local subgroups of

Let G be a finite {p, q}-solvable group for different primes p and q. Let P ∈ Sylp(G) and Q ∈ Sylq(G) be such that PQ = QP . We prove that every χ ∈ Irr(G) of p′-degree has q′-degree if and only if NG(P ) ⊆ NG(Q) and CQ′(P ) = 1.

\pi

Abstract If π is a set of primes, a finite group G is called block π-separated if for every two distinct irreducible complex characters α, β ∈ Irr(G) there is a prime p ∈ π such that α and β are in different p-blocks. The group G is called principally π-separated if the above holds whenever β = 1 G . Bessenrodt and Zhang conjectured that if G is a solvable principally π-separated group then G is π-separated. We construct a family of counter-examples to this conjecture.

-separable groups

Let a finite group A act coprimely on a finite group G. The Glauberman–Isaacs correspondence π(G, A) is a bijection from the set of A-invariant irreducible characters of G onto the set Irr(CG(A)) of irreducible characters of the centralizer of A in G. Let B be a subgroup of A. Composing from left to right, it follows that π(G, A)-1π(G, B) is an injection from Irr(CG(A)) into Irr(CG(B)). We show that, in some cases, the map can be defined via the actions of some subgroups of A containing B on the centralizers in G of some other such subgroups. We also show in many instances, such as |G| odd or A supersolvable and G solvable, that this map is independent of the overgroup G.

Principally separated non-separated solvable groups

1. (i) Suppose K is a conjugacy class of Sn contained in An; then K is called split if K is a union of two conjugacy classes of An. Show that the number of split conjugacy classes contained in An is equal to the number of characters χ ∈ Irr(Sn) such that χAn is not irreducible. (Hint. Consider the vector space of class functions on An which are invariant under conjugation by the transposition (12).)

The quotient of two Glauberman–Isaacs correspondences

If P is a Sylow- p -subgroup of a finite p -solvable group G , we prove that G^\prime \cap \bf{N}_G(P) \subseteq {P} if and only if p divides the degree of every irreducible non-linear p -Brauer character of G. More generally if π is a set of primes containing p and G is π-separable, we give necessary and sufficient group theoretic conditions for the degree of every irreducible non-linear p -Brauer character to be divisible by some prime in π. This can also be applied to degrees of ordinary characters.

Character Theory of Finite Groups

REGULAR EXPRESSIONS . match any single character

VARIATIONS ON THOMPSON'S CHARACTER DEGREE THEOREM

match preceding regular expression at the end of a line ^ match preceding regular expression at the beginning of a line * match zero or more occurrences of preceding expression [ ] match any character in the brackets (or range, i.e. 2-8) [^ ] match any character not in brackets (i.e., ^0-9 means nonnumeric character) \\ last regular expression encountered \(exp\) remember expression for later reference \{m,n\} number of times occurring, with m \{m\} indicating minimum and n \{m,\} indicating maximum