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z-classes in groups: a survey

Sushil Bhunia, Anupam Singh

Abstract

This survey article explores the notion of z-classes in groups. The concept introduced here is related to the notion of orbit types in transformation groups, and types or genus in the representation theory of finite groups of Lie type. Two elements in a group are said to be z-equivalent (or z-conjugate) if their centralizers are conjugate. This is a weaker notion than the conjugacy of elements. In this survey article, we present several known results on this topic and suggest some further questions.

z-classes in groups: a survey

Abstract

This survey article explores the notion of z-classes in groups. The concept introduced here is related to the notion of orbit types in transformation groups, and types or genus in the representation theory of finite groups of Lie type. Two elements in a group are said to be z-equivalent (or z-conjugate) if their centralizers are conjugate. This is a weaker notion than the conjugacy of elements. In this survey article, we present several known results on this topic and suggest some further questions.

Paper Structure

This paper contains 10 sections, 3 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Geometrical motivation and $z$-classes for Lie groups
  3. $z$-classes for classical groups
  4. Counting and types
  5. Classical groups over rings
  6. $z$-classes for algebraic groups
  7. $z$-classes for solvable groups
  8. $z$-classes for finite groups
  9. $z$-classes for finite $p$-groups
  10. $z$-classes for symmetric and alternating groups

Key Result

Theorem 3.4

BS For $q>n$, the number of $z$-classes in $\mathrm{U}_n(q)$ is same as the number of $z$-classes in $\mathrm{GL}_n(q)$. Further, the generating function for the number of $z$-classes is $\prod_{i=1}^{\infty} z(x^i)$, where $z(x)=\prod_{j=1}^{\infty}\frac{1}{(1-x^j)^{p(j)}}$ and $p(j)$ is the number

Theorems & Definitions (15)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Example 1.4
  • Example 2.1
  • Example 3.1
  • Definition 3.2: Fields of type (F)
  • Example 3.3
  • Theorem 3.4
  • Theorem 4.1
  • ...and 5 more