On Some Systems of Collineation Groups

SOME systems of collineation groups which arise in connection with the theory of elliptic functions have been investigated by Klein | and HurwitzJ. One of them is a system in n variables each group of which contains an invariant subgroup of order n. For n SL prime the quotient group with respect to this invariant subgroup is (1, 1) isomorphic with the modular group on two indices of order n(n — 1). The group in three variables is the Hessian group of order 216. For n odd there is also an invariant subgroup of order 2w, and there exist two other groups in (n — l)/2 and (n + l)/2 variables each of which is isomorphic with the quotient group with respect to this subgroup. Thus f or n = 5 there is both a binary and a ternary Cr6o and f ovn = 7 both a ternary and a quaternary GW Similar systems of groups in n, (n — l)/2, and (n + l)/2 variables which arise in the theory of hyperelliptic functions