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.

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 $2$.

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.