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Commutators in Central Products of Cayley-Dickson Loops

Adam Chapman, Ilan Levin

Abstract

This paper studies the triviality of commutators in central products of Cayley-Dickson loops. Two immediate outcomes of this study are (1) the construction of a sequence of non-commutative loops in which the chance of a random commutator to be trivial approaches 1, and (2) an easy proof for why if two central products of $n$-fold Cayley-Dickson loops are isomorphic for $n\geq 3$, then the loops in the first product are term-wise isomorphic to the loops in the second product.

Commutators in Central Products of Cayley-Dickson Loops

Abstract

This paper studies the triviality of commutators in central products of Cayley-Dickson loops. Two immediate outcomes of this study are (1) the construction of a sequence of non-commutative loops in which the chance of a random commutator to be trivial approaches 1, and (2) an easy proof for why if two central products of -fold Cayley-Dickson loops are isomorphic for , then the loops in the first product are term-wise isomorphic to the loops in the second product.
Paper Structure (4 sections, 6 theorems, 4 equations)

This paper contains 4 sections, 6 theorems, 4 equations.

Table of Contents

  1. Cayley-Dickson Algebras
  2. Cayley-Dickson Loops
  3. Commutants and Commutators
  4. Associativity Degree

Key Result

Proposition 3.1

Let $D_1,\dots,D_m$ be $n$-fold Cayley-Dickson loops over $Z$ and let $x \in D_k \setminus Z$ for some $k \in \{1,\dots,m\}$. Then, $C_A(x)/Z$ is of cardinality $2^{(m-1)n+1}=\frac{1}{2^{n-1}}2^{mn}=\frac{1}{2^{n-1}}\cdot|A/Z|$.

Theorems & Definitions (13)

  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • Corollary 3.4
  • proof
  • Remark 3.5
  • Theorem 3.6
  • ...and 3 more