Journal Title/Source
Publicationes Mathematicae Debrecen
Publication Date
2006
Volume
68
Page Numbers
115–137
Document Type
Journal Article
Department
Mathematics and Computer Science
Abstract
C-loops are loops satisfying x(y(yz)) = ((xy)y)z. They often behave analogously to Moufang loops and they are closely related to Steiner triple systems and combinatorics. We initiate the study of C-loops by proving: (i) Steiner loops are C-loops, (ii) C-loops are alternative, inverse property loops with squares in the nucleus, (iii) the nucleus of a C-loop is a normal subgroup, (iv) C-loops modulo their nucleus are Steiner loops, (v) C-loops are power associative, power alternative but not necessarily diassociative, (vi) torsion commutative C-loops are products of torsion abelian groups and torsion commutative 2-C-loops; and several other results. We also give examples of the smallest nonassociative C-loops, and explore the analogy between commutative C-loops and commutative Moufang loops.
Recommended Citation
Phillips, J.D., "C-loops: An introduction" (2006). Journal Articles. 535.
https://commons.nmu.edu/facwork_journalarticles/535