Abstract view
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Degree Homogeneous Subgroups
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Published:2005-03-01
Printed: Mar 2005
John D. Dixon
A. Rahnamai Barghi
Abstract
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.
