Generalized bases of finite groups

Archiv der Mathematik - Motivated by recent results on the minimal base of a permutation group, we introduce a new local invariant attached to arbitrary finite groups. More precisely, a subset...     

Groups whose elements are not conjugate to their powers

We call a finite group irrational if none of its elements is conjugate to a distinct power of itself. We prove that those groups are solvable and describe certain classes of these groups, where the above property is only required for p-elements, for p from a prescribed set of primes.     

Characters and Sylow 2-subgroups of maximal class revisited

We give two ways to distinguish from the character table of a finite group G if a Sylow 2-subgroup of G has maximal class. We also characterize finite groups with Sylow 3-subgroups of order 3 in terms of their principal 3-block.     

Concentration Inequalities for Bounded Functionals via Log-Sobolev-Type Inequalities

Journal of Theoretical Probability - In this paper, we prove multilevel concentration inequalities for bounded functionals $$f = f(X_1, \ldots , X_n)$$ of random variables $$X_1, \ldots , X_n$$...     

Pseudo Frobenius numbers

For a prime $p$, we call a positive integer $n$ a Frobenius $p$-number if there exists a finite group with exactly $n$ subgroups of order $p^a$ for some $a\ge 0$. Extending previous results on Sylow’s theorem, we prove in this paper that every Frobenius $p$-number $n\equiv 1(mod{p}^2)$ is a Sylow $p$-number, i. e., the number of Sylow $p$-subgroups of some finite group. As a consequence, we verify that 46 is a pseudo Frobenius 3-number, that is, no finite group has exactly 46 subgroups of order $3^a$ for any $a\ge 0$.     

Exponent and p-rank of finite p-groups and applications

We bound the order of a finite p-group in terms of its exponent and p-rank. Here the p-rank is the maximal rank of an abelian subgroup. These results are applied to defect groups of p-blocks of finite groups with given Loewy length. Doing so, we improve results in a recent paper by Koshitani, Külshammer, and Sambale. In particular, we determine possible defect groups for blocks with Loewy length 4.     

On the Brauer-Feit bound for abelian defect groups

We improve the Brauer-Feit bound on the number of irreducible characters in a $p$ -block for abelian defect groups by making use of Halasi and Podoski (Every coprime linear group admits a base of size two. http://arxiv.org/abs/1212.0199v1, [7]) and Kessar and Malle (Ann Math 178(2):321–384, [11]). We also prove Brauer’s $k(B)$ -Conjecture for 2-blocks with abelian defect groups of rank at most 5 and 3-blocks and 5-blocks with abelian defect groups of rank at most 3.     

Casimir force on amplifying bodies

Based on a unified approach to macroscopic QED that allows for the inclusion of amplification in a limited space and frequency range, we study the Casimir force as a Lorentz force on an arbitrary partially amplifying system of linearly locally responding (isotropic) magnetoelectric bodies. We demonstrate that the force on a weakly polarisable/magnetisable amplifying object in the presence of a purely absorbing environment can be expressed as a sum over the Casimir-Polder forces on the excited atoms inside the body. As an example, the resonant force between a plate consisting of a dilute gas of excited atoms and a perfect mirror is calculated.     

Orders generated by character values

Let $$K:=\mathbb {Q}(G)$$ be the number field generated by the complex character values of a finite group G. Let $$\mathbb {Z}_K$$ be the ring of integers of K. In this paper we investigate the suborder $$\mathbb {Z}[G]$$ of $$\mathbb {Z}_K$$ generated by the character values of G. We prove that every prime divisor of the order of the finite abelian group $$\mathbb {Z}_K/\mathbb {Z}[G]$$ divides |G|. Moreover, if G is nilpotent, we show that the exponent of $$\mathbb {Z}_K/\mathbb {Z}[G]$$ is a proper divisor of |G| unless $$G=1$$. We conjecture that this holds for arbitrary finite groups G.