Transitive Sets of Homomorphisms

The following remarks were inspired by the discussion in Zassenhaus [l, pp. 51-52] of multiply transitive holomorphs of groups. Theorems 1,2, and 3 below generalize Theorem 6, Theorem 7, and an untheoremed statement, respectively, in [l]. Additional relevant theorems are given in [2]. Let G and H be groups. Let o(G) denote the order of G. Consider the following statements. (A„) o(G)tn+l, o(H)^n+l, and if a^G, x^H, i = l, ■ ■ ■ , », o.-T^Oy, and x^Xj for tVj, a.^Co, and x^es, then there exists a homomorphism a of G into H such that ajo~ = Xi, i= 1, • • • , n. (BB) o(G)^n+l,o(H)^n+l, and iibiGG and ytGH, i=l, n+1, bi5¿bj and yt^y, for if*j, then there exists an h£H and a homomorphism <r of G into H such that A(&,<r) =y,,i=l, ■ ■ • , n + 1. The equivalence of (A„) and (B„) is first proved. Then conditions for the validity of (A„) are investigated. For » = 1 the results are incomplete, but for « ^ 2 they are complete. The proofs are all trivial. We use the notation Q for commutator subgroup, and the term infinitely divisible for a group H such that if hÇ^U and n is a positive integer, then there exists an xÇ.H such that xn = h.