A Class of Loops Categorically Isomorphic to Bruck Loops of Odd Order

Abstract

We define a new variety of loops, Γ-loops. After showing Γ-loops are power-associative, our main goal is showing a categorical isomorphism between Bruck loops of odd order and Γ-loops of odd order. Once this has been established, we can use the well known structure of Bruck loops of odd order to derive the Odd Order, Lagrange and Cauchy Theorems for Γ-loops of odd order, as well as the nontriviality of the center of finite Γ-p-loops (p odd). Finally, we answer a question posed by Jedlička, Kinyon and Vojtěchovský about the existence of Hall π-subloops and Sylow p-subloops in commutative automorphic loops.

Key Words:

• Γ-loops
• Bruck loops
• Power-associativity

2010 Mathematics Subject Classification:

• 20N05

ACKNOWLEDGMENT

I would like to thank Michael Kinyon for suggesting this problem and for his careful reading of each draft of the manuscript. Some investigations in this paper were assisted by the automated deduction tool \textscProver9 [16], the finite model builder \textscMace4 [16], and GAP with the LOOPS package [8, 17].

Notes

Communicated by I. Shestakov.