The geometry of binary systems

G-Loops and Permutation Groups

“…In addition, the autotopy group (see [1,2]), plays a much larger role in loop theory than in group theory. This, and some associated groups and the key maps between them, are displayed in Fig.…”

“…This part of the exposition might be more transparent if done geometrically, from the point of view of 3-nets, as in [1].…”

“…All the relevant definitions are given in Section 2, which stresses the algebraic point of view. These concepts also occur naturally in geometry, since isotopic loops correspond to the same 3-net; see Bruck [2], and Barlotti and Strambach [1].…”

Loops with Transitive Automorphisms

“…Similarly, 1 we can get an example of a loop in class C C by taking the ''column 3 w x adjugate'' of G 11 . This is defined as the quasigroup with multiplication 6 ) given by x и y s z m x ) z s y.…”

“…This implies that the set of autotopisms from 13 with 7 Ž . 1␣s1, namely Aut G , is transitive on the elements not equal to 1,7 whereby G is a tag-loop. Furthermore, for every x g G there is an 7 7 autotopism of G taking 1 to x in the third coordinate.…”