دکتر سجاد محمود رباطی

Let G  be a finite group and let Irr(G)  be the set of irreducible characters of G . ‎We say that an element g  in G  is a vanishing element if there exists some χ∈ Irr(G)   such that χ(g)=0 ‎. In this paper, we provide a relatively short proof for the classification of finite groups whose set of vanishing elements is the :union: of exactly three conjugacy classes‎.

In this paper, we intend to examine the controversy between Frege and Hilbert over geometry about abstract algebra; whether Frege's thinking about abstract algebra works out or Hilbert's? To this end, we first study the times when Frege and Hilbert were engaged in philosophy, to have an idea of the dominant intellectual atmosphere of that time and its influence on these two mathematicians. With this in mind, then we have pointed out some differences between Frege and Hilbert's approaches. We have concluded that their most important differences are the role of the subject in mathematics and the discussion about compatibility. After presenting the normative definition of algebra, we have investigated the points made by Hilbert and Frege, using the later Wittgenstein theories expressed in the books On Certainty and Philosophical Investigation. Finally, by considering all the aspects, we have concluded that what seems to be more useful in the philosophical study of algebra is a third approach, which is Wittgenstein's approach with some modifications, since it guarantees some important features. These features are meaningfulness, intersubjectivity, and rule-orientation, all of which are important features for mathematicians.

Let G be a finite group and let Irr(G) be the set of all irreducible characters of G. We say that an element g in G is a vanishing element if there exists some character χ ∈ Irr(G) such that χ(g) = 0. We can easily show that the set of vanishing elements of G is the union of some conjugacy classes. In this paper, we obtain the set of vanishing elements of Frobenius groups whose kernel is nilpotent of class 2, and then, we will try to find a suitable lower bound for the number of vanishing conjugacy classes of these groups. Moreover, we will classify Frobenius groups containing at most six vanishing conjugacy classes.