The largest character degree and the Sylow subgroups of finite groups (original) (raw)

Character codegrees of maximal class p-groups

arXiv (Cornell University), 2018

Let G be a p-group and let χ be an irreducible character of G. The codegree of χ is given by |G : ker(χ)|/χ(1). If G is a maximal class p-group that is normally monomial or has at most three character degrees then the codegrees of G are consecutive powers of p. If |G| = p n and G has consecutive p-power codegrees up to p n−1 then the nilpotence class of G is at most 2 or G has maximal class.

Bounding an Index by the Largest Character Degree of a p -Solvable Group

Communications in Algebra, 2014

In this paper, we show that if p is a prime and G is a psolvable group, then |G : Op(G)|p ≤ (b(G) p /p) 1/(p−1) where b(G) is the largest character degree of G. If p is an odd prime that is not a Mersenne prime or if the nilpotence class of a Sylow p-subgroup of G is at most p, then |G : Op(G)|p ≤ b(G).

On the Character Degrees of Sylow ppp-subgroups of Chevalley Group of Type E(pf)E(p^f)E(pf)

2010

Let Fq\F_qFq be a field of characteristic ppp with qqq elements. It is known that the degrees of the irreducible characters of the Sylow ppp-subgroup of GLn(Fq)GL_n(\F_q)GLn(Fq) are powers of qqq by Issacs. On the other hand Sangroniz showed that this is true for a Sylow ppp-subgroup of a classical group defined over Fq\F_qFq if and only if ppp is odd. For the classical groups of Lie type BBB, CCC and DDD the only bad prime is 2. For the exceptional groups there are others. In this paper we construct irreducible characters for the Sylow ppp-subgroups of the Chevalley groups D_4(q)D_4(q)D4(q) with q=2fq=2^fq=2f of degree q3/2q^3/2q3/2. Then we use an analogous construction for E6(q)E_6(q)E6(q) with q=3fq=3^fq=3f to obtain characters of degree q7/3q^7/3q7/3, and for E8(q)E_8(q)E_8(q) with q=5fq=5^fq=5f to obtain characters of degree q16/5.q^{16}/5.q16/5. This helps to explain why the primes 2, 3 and 5 are bad for the Chevalley groups of type EEE in terms of the representation theory of the Sylow ppp-subgroup.

p-Parts of Brauer character degrees

Journal of Algebra, 2014

Let G be a finite group and let p be an odd prime. Under certain conditions on the p-parts of the degrees of its irreducible p-Brauer characters, we prove the solvability of G. As a consequence, we answer a question proposed by B. Huppert in 1991: If G has exactly two distinct irreducible p-Brauer character degrees, then G is solvable. We also determine the structure of non-solvable groups with exactly two irreducible 2-Brauer character degrees.

Groups with exactly one irreducible character of degree divisible by p

Algebra & Number Theory, 2014

Let p be a prime. We characterize those finite groups which have precisely one irreducible character of degree divisible by p. Minimal situations constitute a classical theme in group theory. Not only do they arise naturally, but they also provide valuable hints in searching for general patterns. In this paper, we are concerned with character degrees. One of the key results on character degrees is the Itô-Michler theorem, which asserts that a prime p does not divide the degree of any complex irreducible character of a finite group G if and only if G has a normal, abelian Sylow p-subgroup. In ], Isaacs together with the fourth, fifth, and sixth authors of this paper studied the finite groups that have only one character degree divisible by p. They proved, among other things, that the Sylow p-subgroups of those groups were metabelian. This suggested that the derived length of the Sylow p-subgroups might be related with the number of different character degrees divisible by p. However, nothing could be said in ] on how large p-Sylow normalizers were inside G. (As a trivial example, the dihedral group of order 2n for n odd has a unique character degree divisible by 2, and a self-normalizing Sylow 2-subgroup of order 2.) In this paper, we go further and completely classify the finite groups with exactly one irreducible character of degree divisible by p. Our focus now therefore is not only on the set of character degrees but also on the multiplicity of the number of irreducible characters of each degree. In Section 1, we define the terms semiextraspecial, ultraspecial, and doubly transitive Frobenius groups of Dickson type. We would like to thank the referee for the helpful suggestions. Guralnick and Tiep gratefully acknowledge the support of the NSF (grants DMS-1001962, DMS-0901241, and DMS-1201374).

Constructing characters of Sylow p-subgroups of finite Chevalley groups

Let q be a power of a prime p, let G be a finite Chevalley group over F q and let U be a Sylow p-subgroup of G; we assume that p is not a very bad prime for G. We explain a procedure of reduction of irreducible complex characters of U , which leads to an algorithm whose goal is to obtain a parametrization of the irreducible characters of U along with a means to construct these characters as induced characters. A focus in this paper is determining the parametrization when G is of type F 4 , where we observe that the parametrization is "uniform" over good primes p > 3, but differs for the bad prime p = 3. We also explain how it has been applied for all groups of rank 4 or less.

Characters of the Sylow p-subgroups of the Chevalley groups

Journal of Algebra, 2011

Let U (q) be a Sylow p-subgroup of the Chevalley groups D 4 (q) where q is a power of a prime p. We describe a construction of all complex irreducible characters of U (q) and obtain a classification of these irreducible characters via the root subgroups which are contained in the center of these characters. Furthermore, we show that the multiplicities of the degrees of these irreducible characters are given by polynomials in q − 1 with nonnegative coefficients.