On The Isomorphism Classes Of Transversals III

TL;DR

This paper develops a method to count the number of isomorphism classes of transversals in finite groups, which correspond to non-isomorphic left loops of a given order, advancing understanding of algebraic structures related to groups.

Contribution

It introduces a new method for calculating the number of isomorphism classes of transversals and non-isomorphic left loops in finite groups.

Findings

Derived a formula for counting isomorphism classes of transversals.

Calculated the number of non-isomorphic left loops of specific orders.

Provided a systematic approach to classify algebraic structures related to group transversals.

Abstract

Let $G$ be a finite group and $H$ a subgroup of $G$. Each left transversal (with identity) of $H$ in $G$ has a left loop (left quasigroup with identity) structure induced by the binary operation of $G$. We say two left transversals are isomorphic if they are isomorphic with respect to the induced left loop structures. In this paper, we develop a method to calculate the number of isomorphism classes of transversals of $H$ in $G$. Also with the help of this we calculate the number of non-isomorphic left loops of a given order.