Asian Journal of Pure and Applied Mathematics

Investigation on Lesser Prime Order Subloops of Odd Order Moufang Loops

Lois A. Ademola *

Department of Mathematics, University of Jos, Jos, Nigeria.

Garba G. Zaku

Department of Mathematics, University of Jos, Jos, Nigeria.

*Author to whom correspondence should be addressed.

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Abstract

Ademola and Rajah have proven that if L is a moufang loop of odd order p^α m, where p is the smallest prime dividing |L| with (p,m)=1 and α∈{1,2}. Then there exists a subloop L_m of order m normal in L. This paper goes further to investigate the existence of a normal subloop for Moufang loop of any odd order, with the condition that every proper subloop and quotient loop of L is a group. This is done by considering a subloop L₁ of order p^α q^β where p and q are odd primes with 3<p<q, q ≢ ±1(mod p), α ∈Z⁺ and 1 ≤ β ≤ 2, and investigating the possible existence of a normal p - subloop of order p^α in L₁ .

Keywords: Natural pests, Moufang loop, Disease resistance, order, prime, nonassociative, normal, subloops

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