arXiv subjects
Publications and source records attributed to Christopher Herbig.
A group $H$ is said to be a $B$-group if whenever a primitive permutation group $G$ contains a regular subgroup isomorphic to $H$, then $G$ is doubly transitive. Going as far back as William Burnside, several authors have investigated whether certain families of groups are $B$-groups. Using the O'Nan-Scott Theorem as our starting point, we classify the $B$-groups of order $p^4$ where $p$ is a prime.
Let $\chi$ be a complex irreducible character of a finite group $G$. The conductor of $\chi$, denoted $c(\chi)$, is the smallest positive integer $n$ such that $\chi(x)\in \mathbb{Q}(\exp({2\pi i/n}))$ for all $x\in G$. We show that for certain rank $1$ finite groups of Lie type, the conductor $c(\chi)$ is realized at a single group element; that is, there exists $g\in G$ such that $c(\chi)=c(\chi(g))$. In some quasisimple cases, we further prove that the field of values \(\mathbb{Q}(\chi)\) is generated by a single value. This phenomenon, which is related to a well-known conjecture of W.~Feit, was recently observed by Boltje \emph{et al.} in their reduction of the conjecture to finite simple groups. Our approach uses techniques from algebraic number theory together with the known character tables of these groups.
A character of a finite group having degree $n$ takes values which may be expressed as sums of $n$ or fewer roots of unity. In this note, we prove a result which describes the irreducible constituents of generalized characters on abelian groups whose values on nonidentity elements are expressible as sums of two or fewer roots of unity. In Section 4, we apply our main result to obtain information about the connectivity of prime graphs for groups admitting such characters.
In Question 5.2 of [5], Hung and Tiep asked the following: If $\alpha$ is a sum of $k$ complex roots of unity and $\mathbb{Q}_{c(\alpha)}$ is the smallest cyclotomic field containing $\alpha$, is it true that $|\mathbb{Q}_{c(\alpha)}:\mathbb{Q}(\alpha)| \leq k$? We answer this question in the negative. Using known results on minimal vanishing sums, we also characterize all cyclotomic integers with $k \leq 4$ for which the inequality fails. In \S 6, we bound the growth of $|\mathbb{Q}_{c(\alpha)}:\mathbb{Q}(\alpha)|$ as a function of $k$.
In [3, Theorem 6.7B], the authors use the Main Theorems of Brauer to give a proof of Burnside's Normal $p$-complement Theorem. Unfortunately, the proof contains an error. We take this opportunity to give a proof along similar lines, circumventing the error by means of a well-known result on traces of totally positive cyclotomic integers.