Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Compact $p$-adic analytic groups in which centralizers are abelian

Luis Mendonça, Thomas S. Weigel, Theo Zapata

Abstract

Using methods of associative algebras, Lie theory, group cohomology, and modular representation theory, we construct profinite $p$-adic analytic groups such that the centralizer of each of their non-trivial elements is abelian. The paper answers questions of P.~Shumyatsky, P.~Zalesskii, and T.~Zapata in the Israel J. Math., v.~230, 2019.

Compact $p$-adic analytic groups in which centralizers are abelian

Abstract

Using methods of associative algebras, Lie theory, group cohomology, and modular representation theory, we construct profinite -adic analytic groups such that the centralizer of each of their non-trivial elements is abelian. The paper answers questions of P.~Shumyatsky, P.~Zalesskii, and T.~Zapata in the Israel J. Math., v.~230, 2019.

Paper Structure

This paper contains 18 sections, 25 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Lie algebras
  3. Solvable Lie algebras
  4. Simple Lie algebras
  5. Proof of Theorem \ref{['t:CTLieAlgebras']} and its corollaries
  6. $p$-adic Lie groups with CA Lie algebras
  7. Some universal $p$-Frattini extensions
  8. Group extensions with abelian kernel
  9. Properties of the universal $p$-Frattini cover with abelian kernel
  10. The universal $p$-Frattini extension with elementary abelian kernel
  11. Central involutions
  12. The $(-1)$-universal $2$-Frattini extension with abelian kernel
  13. Ranks
  14. $\mathrm{SL}_2$-like finite groups with the $\Omega^\circ$-fixed point property
  15. $\mathrm{SL}_2$-like finite groups
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $D$ be a central division algebra of prime degree $\ell$ over $\mathbb{Q}_p$. If $p$ is odd and $\ell$ does not divide $p-1$, then the profinite $p$-adic analytic group $\mathrm{SL}_1(D)$ is a CSA-group which is neither pro-$p$ nor abelian.

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Lemma 2.1: KlMo10
  • Lemma 2.2: KlMo10
  • Proposition 2.3
  • ...and 34 more