English

Finite Bruck Loops

Group Theory 2008-01-15 v1

Abstract

A loop (X,)(X,\circ) is said to be a Bruck loop if it satisfies the (right) Bol identity ((zx)y)x=z((xy)x)((z\circ x)\circ y)\circ x = z\circ ((x\circ y)\circ x) and the automorphic inverse property (xy)1=x1y1(x\circ y)^{-1}=x^{-1}\circ y^{-1}. If XX is a finite Bruck loop and GG is the group generated by all right translations R(x):yyxR(x): y\mapsto y\circ x, then we show that XX and GG are central products X=O2(X)O(X)X = O^{2'}(X) * O(X) and G=O2(G)O(G)G = O^{2'}(G) * O(G), where O2(X)O^{2'}(X) (O2(G)O^{2'}(G)) is the subloop (subgroup) generated by all 2-elements, and O(X)O(X) (O(G)O(G)) is the largest normal subloop (subgroup) of odd order. In particular, if XX is solvable, then these central products are direct products. We also give a set of necessary conditions that must hold for a finite Bruck loop XX to be nonsolvable but have each proper section solvable; in particular, XX must be simple and consist of 2-elements, while the quotient of GG by its largest normal 2-subgroup must be isomorphic to PGL2(q)PGL_2(q), with q=2n+15q=2^n+1\geq 5.

Keywords

Cite

@article{arxiv.math/0401193,
  title  = {Finite Bruck Loops},
  author = {Michael Aschbacher and Michael K. Kinyon and J. D. Phillips},
  journal= {arXiv preprint arXiv:math/0401193},
  year   = {2008}
}

Comments

16 pages, AMS-TeX