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On Holomorph of WIP PACC Loops

Olufemi. O. George

Department of Mathematics, University of Lagos, Akoka, Nigeria

Email address: oogeorge@unilag.edu.ng

Doi : https://doi.org/10.47013/16.3.6

Cited by : Jordan J. Math & Stat., 16 (3) (2023), 463 - 482

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Received on: June 28, 2022;                                         Accepted on: Jan. 25, 2023

 Abstract. This work investigates 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 ϵ 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 ϵ NÁ(Q), then the holomorph of Q is an extra loop.

Keywords: left (right) conjugacy closed loop, power associative, weak inverse property, LWPC-loop, RWPC-loop, WIP PACC loop, holomorph.