On the non-occurrence of the coxeter graphs β<SUB>2n+1</SUB>, D<SUB>2n+1</SUB> and E<SUB>7</SUB> as the principal graph of an inclusion of II<SUB>1</SUB> factors

After discussing some preliminaries on the notion of an action of a hypergroup on a set, we present elementary proofs of the fact that the coxeter graphs β2n+1 D2n+1 and E7 do not arise as Jones' principal graph invariant of an inclusion of II1 factors. (Here, we use the symbol an to denote the graph that is normally denoted by Bn, the reason for this changed terminology being spelt out in the text.