Geodesic
A class of commutative loops with metacyclic inner mapping groups
Commentationes Mathematicae Universitatis Carolinae, Tome 49 (2008) no. 3, pp. 357-382.

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We investigate loops defined upon the product $\Bbb Z_m\times \Bbb Z_k$ by the formula $(a,i)(b,j) = ((a+b)/(1+tf^i(0)f^j(0)), i + j)$, where $f(x) = (sx + 1)/(tx+1)$, for appropriate parameters $s,t \in \Bbb Z_m^*$. Each such loop is coupled to a 2-cocycle (in the group-theoretical sense) and this connection makes it possible to prove that the loop possesses a metacyclic inner mapping group. If $s=1$, then the loop is an A-loop. Questions of isotopism and isomorphism are considered in detail.
Classification : 08A05, 20N05
Keywords: A-loop; nucleus; inner mapping group; cocycle; linear fractional 
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     title = {A class of commutative loops with metacyclic inner mapping groups},
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Drápal, Aleš. A class of commutative loops with metacyclic inner mapping groups. Commentationes Mathematicae Universitatis Carolinae, Tome 49 (2008) no. 3, pp. 357-382. http://geodesic.mathdoc.fr/item/CMUC_2008__49_3_a0/