On the commutative center of Moufang loops

TL;DR

This paper constructs infinite series of Moufang loops of exponent 3 with non-normal commutative centers, providing explicit examples of such loops of specific orders and linking one to the Moufang triplication of a free Burnside group.

Contribution

It introduces new examples of Moufang loops with non-normal commutative centers, including the first known loops of certain orders with this property.

Findings

Constructed Moufang loops of orders 3^8 and 3^{11} with non-normal commutative centers.

Demonstrated that the commutative center need not be a normal subloop in Moufang loops.

Connected one example to the Moufang triplication of a free Burnside group B(3,3).

Abstract

We construct two infinite series of Moufang loops of exponent $3$ whose commutative center (i.e. the set of elements that commute with all elements of the loop) is not a normal subloop. In particular, we obtain examples of such loops of orders $3^8$ and $3^{11}$ one of which can be defined as the Moufang triplication of the free Burnside group $B(3,3)$.