Unbreakable Loops

TL;DR

This paper introduces the concept of unbreakable loops, nonassociative structures with no nontrivial subloops, and constructs infinite families of such loops with specific symmetry properties for various orders.

Contribution

It demonstrates the existence of nonassociative unbreakable loops for all orders n >= 5 and describes their multiplication groups, including cases where these groups are symmetric or alternating groups.

Findings

Unbreakable loops exist for all orders n >= 5.

Constructed families of commutative unbreakable loops with specific multiplication groups.

Existence of unbreakable loops with symmetric group multiplication for even n >= 6.

Abstract

We say that a loop is unbreakable when it does not have nontrivial subloops. While the cyclic groups of prime order are the only unbreakable finite groups, we show that nonassociative unbreakable loops exist for every order n >= 5. We describe two families of commutative unbreakable loops of odd order, n >= 7, one where the loop's multiplication group is isomorphic to the alternating group An and another where the multiplication group is isomorphic to the symmetric group Sn. We also prove for each even n >= 6 that there exist unbreakable loops of order n whose multiplication group is isomorphic to Sn.