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Splitting of Clifford groups associated to finite abelian groups

César Galindo

Abstract

The Clifford group associated with a finite abelian group gives rise to a natural extension by the corresponding symplectic group. We prove that this extension splits as a semidirect product if and only if the group order is not divisible by four. This confirms a conjecture of Korbelář and Tolar and extends their cyclic result to arbitrary finite abelian groups.

Splitting of Clifford groups associated to finite abelian groups

Abstract

The Clifford group associated with a finite abelian group gives rise to a natural extension by the corresponding symplectic group. We prove that this extension splits as a semidirect product if and only if the group order is not divisible by four. This confirms a conjecture of Korbelář and Tolar and extends their cyclic result to arbitrary finite abelian groups.

Paper Structure

This paper contains 19 sections, 11 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: cohomology and the Clifford extension
  3. Group extensions and factor sets
  4. Schur Multiplier of finite abelian groups
  5. The Heisenberg extension
  6. The pseudo-symplectic group Ps(A) and the Clifford group
  7. The exact sequence and the obstruction cocycle
  8. Decomposition and splitting reductions
  9. Coprime decomposition
  10. Splitting at odd primes
  11. Reduction to direct summands
  12. Non-splitting for Z_2k̂
  13. Explicit lifts of the generators
  14. Incompatible lifting constraints
  15. Symplectic obstruction in the elementary abelian case
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $A$ be a finite abelian group. The Clifford extension eq:extension splits as a semidirect product if and only if $4 \nmid |A|$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Theorem 3.3
  • proof
  • Lemma 4.1
  • ...and 12 more