2012/06/06 by Mark Greer, Greer, Mark
Mathematics · #20N05 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20N05
paper · pdf · doi:10.48550/arxiv.1206.1356
arxiv created 2013/02/09 · arxiv updated 2013/02/12
We define a new variety of loops we call Γ-loops. After showing Γ-loops are power associative, our main goal will be 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. By showing commutative automorphic loops are Γ-loops and using the categorical isomorphism, we answer in the affirmative.