On Smarandache Bryant Schneider group of a Smarandache loop

TL;DR

This paper introduces the Smarandache Bryant Schneider Group for Smarandache loops, explores its properties and relationships with other groups, and provides formulas for finite cases, advancing the understanding of loop isotopy and isomorphy.

Contribution

It defines the Smarandache Bryant Schneider Group, investigates its properties and relationships with Bryant Schneider Groups, and derives cardinality formulas for finite Smarandache loops.

Findings

Relationship between Bryant Schneider Group and Smarandache Bryant Schneider Group established

Properties of Bryant Schneider Group extended to Smarandache loops

Cardinality formulas for finite Smarandache loops derived

Abstract

The concept of Smarandache Bryant Schneider Group of a Smarandache loop is introduced. Relationship(s) between the Bryant Schneider Group and the Smarandache Bryant Schneider Group of an S-loop are discovered and the later is found to be useful in finding Smarandache isotopy-isomorphy condition(s) in S-loops just like the formal is useful in finding isotopy-isomorphy condition(s) in loops. Some properties of the Bryant Schneider Group of a loop are shown to be true for the Smarandache Bryant Schneider Group of a Smarandache loop. Some interesting and useful cardinality formulas are also established for a type of finite Smarandache loop.