A Characterization of Group Through Isomorphism Classes of Transversals

TL;DR

This paper characterizes groups based on the number of isomorphism classes of right transversals, establishing a specific link between the number of classes, subgroup index, and permutation representations.

Contribution

It proves that having exactly five isomorphism classes of right transversals implies a subgroup index of six and a permutation representation isomorphic to A4.

Findings

Number of isomorphism classes of transversals is 5

Subgroup index in G is 6

Permutation representation is isomorphic to A4

Abstract

Let G be a group and H a subgroup of G of finite index. In this article, it is proved that if the number of isomorphism classes of right transversals of H in G is 5, then the index of H in G is 6 and the permutation representation of G on right cosets of H in G is isomorphic to the alternating group on four symbols.