On the dimensions of automorphism groups of 8-dimensional ternary fields. I (Q1346997)

It is known that the dimension of a finite-dimensional, locally compact, connected ternary field is equal to 1, 2, 4, or 8. By results of H. Salzmann, every 1-dimensional ternary field is rigid and a 2-dimensional ternary field admits at most 2 continuous automorphisms. Recently, the author has proved [Math. Z. 215, No. 1, 89-97 (1994; Zbl 0809.22005)] that the automorphism group of a 4-dimensional ternary field is at most 4-dimensional. In the paper under review, he proves similar results for 8-dimensional ternary fields. In this case, the automorphism group either is isomorphic to the compact exceptional Lie group of type \(G_2\) or it is at most 11-dimensional. If the ternary field fixed by the automorphism group is connected this bound can be replaced by 10. Similar result also holds for the more general case of locally compact double loops, cf. the author [Monatsh. Math. 117, No. 1-2, 1-16 (1994; Zbl 0796.22004)].