Extensions of the Klein-4 group and signed products [Articol]

Abstract

Signed superquasigroups allow the creation of a (signed) super product for quasigroups similar to that which exists for superalgebras. They also permit the creation of the quatedral loop, a nonassociative loop with the same character table as the groups D4 and Q8. However, this loop was not created by directly taking the signed superproduct, but required some additional work. We show that it is impossible to create the quatedral loop as a signed superproduct of signed superquasigroups. In fact, all signed superproducts of order 4 signed superquasigroups are isotopic to Q8 or D4. A related result on signed superproducts applies to signed superproducts of an entire class of signed superquasigroups called standard superquasigroups: signed superquasigroups which are extensions of superloops by Z=2. Signed superproducts of such loops have particularly nice properties. A particularly interesting example of standard loops are hedral loops, which are extensions of Z=2 Z=2 by Z=2. The character tables of such loops are examined, and shown to share the character table of Q8 or that of a non-cyclic abelian group of order 8. Finally, results on hedral loops are used to show that the 16 element octonion loop is not isotopic to a signed product of standard signed superquasigroups.