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Schur covers of skew braces

T. Letourmy, L. Vendramin

Abstract

We develop the theory of Schur covers of finite skew braces. We prove the existence of at least one Schur cover. We also compute several examples. We prove that different Schur covers are isoclinic. Finally, we prove that Schur covers have the lifting property concerning projective representations of skew braces.

Schur covers of skew braces

Abstract

We develop the theory of Schur covers of finite skew braces. We prove the existence of at least one Schur cover. We also compute several examples. We prove that different Schur covers are isoclinic. Finally, we prove that Schur covers have the lifting property concerning projective representations of skew braces.
Paper Structure (15 sections, 59 theorems, 205 equations)

This paper contains 15 sections, 59 theorems, 205 equations.

Table of Contents

  1. Preliminaries
  2. Group cohomology
  3. Annihilator extensions
  4. Cohomology of trivial and almost trivial skew braces
  5. Hoschild--Serre exact sequence for skew braces
  6. The Schur multiplier
  7. Trivial and almost trivial skew braces
  8. Cyclic skew braces
  9. Direct product
  10. Direct product of some bicyclic skew braces
  11. Skew braces of order $p^2$
  12. Schur covers
  13. The relation between $M_b(Q)$ and $H_b^2(Q,A)$
  14. Some examples
  15. Representation Theory

Key Result

Lemma 1.6

Let $N$ and $T$ be finite groups.

Theorems & Definitions (149)

  • Example 1.1
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • Lemma 1.6
  • proof
  • Theorem 1.7
  • proof
  • Definition 1.8
  • Definition 1.9
  • ...and 139 more