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

Abstract

We define a new variety of loops, -loops. After showing -loops are power-associative, our main goal is showing a categorical isomorphism between Bruck loops of odd order and -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 -loops of odd order, as well as the nontriviality of the center of finite -p-loops (p odd). Finally, we answer a question posed by Jedlička, Kinyon and Vojtěchovský about the existence of Hall -subloops and Sylow p-subloops in commutative automorphic loops.

Enumeration of involutory latin quandles, Bruck loops and commutative automorphic loops of odd prime power order

“…Let p be an odd prime. In [14], Greer asked if the Γ-loops associated with left Bruck loops of order p 3 are always commutative automorphic loops. We can generalize his question as follows: Problem 8.1.…”

“…Note that the correspondence of Theorem 3.1 is vacuous when |Q| is even. The following correspondence was proved in [14]: Theorem 3.2 (Greer). There is a one-to-one correspondence between left Bruck loops of odd order n and Γ-loops of odd order n. In more detail:…”

“…Like Bruck loops, commutative automorphic loops of odd prime power order are centrally nilpotent [18]. Moreover, there is a one-to-one correspondence between Bruck loops of odd order and the so-called Γ-loops of odd order [14], a class of loops that contains all commutative automorphic loops of odd order.…”

“…It was known that there exist Γ-loops of odd order that are not commutative automorphic loops [14], but it was not known until now if there exists a Γ-loop of odd prime power order that is not a commutative automorphic loop. By exhaustively inspecting the Bruck loops obtained in our enumeration, we found several Bruck loops of order 3 5 whose corresponding Γ-loops are not commutative automorphic loops.…”

Automorphic loops and metabelian groups

“…We investigate some general properties and applications of • and determine a necessary and sufficient condition on G in order for (G, •) to be Moufang. In [6], it is conjectured that G is metabelian if and only if (G, •) is an automorphic loop. We answer a portion of this conjecture in the affirmative: in particular, we show that if G is a split metabelian group of odd order, then (G, •) is automorphic.…”