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1. CMB 2013 (vol 57 pp. 449)
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)
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)
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 $2$.
Category:20C15 |
4. CMB 2005 (vol 48 pp. 41)
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.
Category:20C15 |