On The Inductor, Restrictor Of A Character Pair(Triple) And Their Linear Reductions

The representation theory of finite groups is an important branch of modern mathematics, and the character theory is one of the main tools to study the ordinary representation theory. Character pairs(triples) are one of the most basic research objects in character theory, and they play an important role in the studying of both group theory and character theory .The inductor, restrictor of a character pair (triple) and its linear reduction are further discussed in this paper. In the "Introduction", we have presented the backgrounds, research orientations and development trends related to this paper. In chapter 1, some basic concepts and important conclusions are given so that they can lay a massive foundation for the next two chapters. In chapter 2, we have extended the results of Isaacs's in [4] by proving that there exists a middle character triple between a character triple and its inductor (restrictor), and that a special map between the related groups and a bijection between the related irreducible characters can be found. In chapter 3, we have explored that in some situations we can use the same linear character to linearly reduce a character pair (triple) and its inductor(restrictor), and that "Induction"("Restriction") also holds between the reduced character triples.