The commutative Moufang loops with minimum conditions for subloops I

TL;DR

This paper investigates commutative Moufang loops with minimal subloop conditions, revealing their structure as finite extensions of quasicyclic groups and establishing equivalences between various finiteness conditions.

Contribution

It characterizes the structure of CMLs with minimal subloop conditions and proves the equivalence of these conditions with finiteness properties of their multiplication groups.

Findings

CMLs with minimum subloop conditions are finite extensions of quasicyclic groups.

The minimum conditions for subloops and normal subloops are equivalent in CMLs.

Such CMLs are characterized by finiteness conditions of their multiplication groups.

Abstract

The structure of the commutative Moufang loops (CML) with minimum condition for subloops is examined. In particular it is proved that such a CML $Q$ is a finite extension of a direct product of a finite number of the quasicyclic groups, lying in the centre of the CML $Q$. It is shown that the minimum conditions for subloops and for normal subloops are equivalent in a CML. Moreover, such CML also characterized by different conditions of finiteness of its multiplication groups.