ON LOOPS OF VESANEN TYPE OF SMALL ORDERS

ON LOOPS OF VESANEN

Abstract

 Bruck in $1946$ showed that if $Q$ is a nilpotent loop
with $CL(Q)\leq 2$, then $Inn(Q)$ is abelian. However, Vesanen example in
\cite{mon50} showed a nilpotent loop $Q$ of order $18$ with $CL(Q)= 3$
such that the $Inn(Q)$ is not even nilpotent. These type of loops are
called, in this paper, loops of Vesanen type. Examples and properties of these type of
loops are shown and examined. The results generally showed that a loop of Vesanen type is centrally nilpotent
and can either be abelian or non-abelian but with non-nilpotent inner
mapping group. The characterization of these loops are obtained and presented.