A study of the holomorphy of (r, s, t)-inverse loops

Abstract

In this paper, the holomorphic study of inverse properties in loops is put in a more general setting. For various combinations of r, s, t ∈ ℤ in parities, it was established that: (i) the A(Q)-holomorph H(Q) of a loop Q is an (r, s, t)-inverse loop if and only if Q is an (r, s, t) -inverse loop; (ii) the A(Q) -holomorph H(Q) of a loop Q is an (r, s, t)-inverse loop if and only if Q is an (r, s, t)-inverse loop, A(Q) is a particular kind of group (e.g abelian, Boolean) and any two elements of A(Q) satisfies some autotopic conditions. Specifically, m-inverse loop (when m is odd), double weak inverse property loop (WWIPL) and weak inverse property loop were found to satisfy the case (i) while m-inverse loop (when m is even) and weak inverse property loop were found to satisfy case (ii). For a Buchsteiner loop (which is a special kind of WWIPL) Q, it was shown that the A(Q)-holomorph H(Q) is a Buchsteiner loop if and only if A(Q) is a nuclear automorphism group. The left (right) inner automorphism group of a Buchsteiner loop Q was shown to be a normal subgroup of the automorphism group of Q. Existing examples of loops which are relevant to this study were cited.