On the translation functors for a semisimple algebraic group.
Let be a finite group and the cyclic group of order 2. Consider the 8 multiplicative operations , where . Define a new multiplication on by assigning one of the above 8 multiplications to each quarter , for . If the resulting quasigroup is a Bol loop, it is Moufang. When is nonabelian then exactly four assignments yield Moufang loops that are not associative; all (anti)isomorphic, known as loops .
We give the characterization of the unit group of , where is a finite field with elements for prime and denotes the special linear group of matrices having determinant over the cyclic group .