We are interested in determining the bound of the average of the degrees of the irreducible characters whose degrees are not divisible by some prime $p$ that guarantees a finite group $G$ of odd order is $p$-nilpotent. There was found to be a bound which depends on the prime $p$. By further restricting the average by fixing a subfield $k$ of the complex numbers and then computing the average of the degrees of the irreducible characters whose degrees are not divisible by $p$ and have values in $k$, then a bound was obtained which depends on both $p$ and $k$. Examples are provided which show best possible bounds.

Let $f(x) \in \mathbb{Z}[x]$ be monic with $\mathrm{deg}(f) = N \geq 2$. Suppose that $f(x)$ is monogenic, and that $f(x)$ is the characteristic polynomial of the $N$th order linear recurrence sequence $\Upsilon_f := (U_n)_{n \geq 0}$ with initial conditions $$U_0 = U_1 = \cdots = U_{N−2} = 0 \mathrm{\hspace {1 pc}and\hspace {1 pc}} U_{N−1} = 1.$$ Let $\pi (m)$ denote the length of the period of $\Upsilon_f$ modulo the integer $m \geq 2$, where $\mathrm{gcd}(m, f(0)) = 1$. Let $p$ be a prime such that $f(x)$ is irreducible over $\mathbb{F}_p$ and $f(x^p)$ is irreducible over $\mathbb{Q}$. We prove that $f(x^p)$ is monogenic if and only if $\pi (p^2) \neq \pi(p)$, which provides a new and simple test for the monogenicity of $f(x^p)$. We also present some infinite families of such polynomials. This article extends previous work of the author.

For a quadratic module, degenerate or not, over a commutative ring with 1, we determine some of the important subalgebras of its Clifford algebra under some conditions, and consider the Clifford group and related ones with the maps to the orthogonal group of the module. By allowing elements that are not necessarily invertible, but only have a property that we call pseudo-invertibility, we extend these groups to larger monoids, which still carry natural maps to the orthogonal group, with the image sometimes containing orthogonal transformations that are not in the image of any group from the classical theory. Our objects may contain elements that are not locally homogeneous, which allows us obtain an analogous theory in the paravector setting.