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Search: MSC category 20C15 ( Ordinary representations and characters )

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1. CMB 2013 (vol 57 pp. 449)

Alaghmandan, Mahmood; Choi, Yemon; Samei, Ebrahim
ZL-amenability Constants of Finite Groups with Two Character Degrees
We calculate the exact amenability constant of the centre of $\ell^1(G)$ when $G$ is one of the following classes of finite group: dihedral; extraspecial; or Frobenius with abelian complement and kernel. This is done using a formula which applies to all finite groups with two character degrees. In passing, we answer in the negative a question raised in work of the third author with Azimifard and Spronk (J. Funct. Anal. 2009).

Keywords:center of group algebras, characters, character degrees, amenability constant, Frobenius group, extraspecial groups
Categories:43A20, 20C15

2. CMB 2006 (vol 49 pp. 285)

Riedl, Jeffrey M.
Orbits and Stabilizers for Solvable Linear Groups
We extend a result of Noritzsch, which describes the orbit sizes in the action of a Frobenius group $G$ on a finite vector space $V$ under certain conditions, to a more general class of finite solvable groups $G$. This result has applications in computing irreducible character degrees of finite groups. Another application, proved here, is a result concerning the structure of certain groups with few complex irreducible character degrees.

Categories:20B99, 20C15, 20C20

3. CMB 2006 (vol 49 pp. 127)

Lewis, Mark L.
Character Degree Graphs of Solvable Groups of Fitting Height $2$
Given a finite group $G$, we attach to the character degrees of $G$ a graph whose vertex set is the set of primes dividing the degrees of irreducible characters of $G$, and with an edge between $p$ and $q$ if $pq$ divides the degree of some irreducible character of $G$. In this paper, we describe which graphs occur when $G$ is a solvable group of Fitting height $2$.


4. CMB 2005 (vol 48 pp. 41)

Dixon, John D.; Barghi, A. Rahnamai
Degree Homogeneous Subgroups
Let $G$ be a finite group and $H$ be a subgroup. We say that $H$ is \emph{degree homogeneous }if, for each $\chi\in \Irr(G)$, all the irreducible constituents of the restriction $\chi_{H}$ have the same degree. Subgroups which are either normal or abelian are obvious examples of degree homogeneous subgroups. Following a question by E.~M. Zhmud', we investigate general properties of such subgroups. It appears unlikely that degree homogeneous subgroups can be characterized entirely by abstract group properties, but we provide mixed criteria (involving both group structure and character properties) which are both necessary and sufficient. For example, $H$ is degree homogeneous in $G$ if and only if the derived subgroup $H^{\prime}$ is normal in $G$ and, for every pair $\alpha,\beta$ of irreducible $G$-conjugate characters of $H^{\prime}$, all irreducible constituents of $\alpha^{H}$ and $\beta^{H}$ have the same degree.


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