On Holomorph of WIP PACC Loops

Abstract

This workinvestigates the holomorph of a weak inverse property power associative conjugacy closed (WIP PACC) loop. It is shown that the holomorph of a WIP PACC loop is WIP PACC. If Q is a WIP PACC loop and A is the automorphism group of Q, then each θ ∈ A is a nuclear automorphism. The A(Q) holomorph of a WIP PACC loop is shown to satisfy the doubly weak inverse property. A necessary and sufficient condition for the holomorph of an arbitrary loop and its automorphism group to produce a WIP PACC loop is established.
Finally, if Q is a LWPC (RWPC) loop with x ∈ Nµ(Q), then the holomorph of Q is an extra loop.

Key words and phrases. left (right) conjugacy closed loop, power associative, weak inverse property, LWPC-loop, RWPC-loop, WIP PACC loop, holomorph.

2010 Mathematics Subject Classification. 20N02, 20N05