Generalization of the basis theorem for alternating groups

TL;DR

This paper extends the concept of ordered generating systems (OGS) from symmetric groups to their alternating subgroups, providing new insights into their structure and properties.

Contribution

It introduces a generalized OGS for alternating groups, along with exchange laws and properties, expanding the understanding of group presentations.

Findings

Generalized OGS for alternating groups defined

Exchange laws for the alternating subgroup established

Properties of the alternating subgroup presentation analyzed

Abstract

There were defined by R. Shwartz OGS for non-abelian groups, as an interesting generalization of the basis of finite abelian groups. The definition of OGS states that that every element of a group has a unique presentation as a product of some powers of the OGS, in a specific given order. In case of the symmetric groups S_{n} there is a paper of R. Shwartz, which demonstrates a strong connection between the OGS and the standard Coxeter presentation of the symmetric group. The OGS presentation helps us to find the Coxeter length and the descent set of an arbitrary element of the symmetric group. Therefore, it motivates us to generalize the OGS for the alternating subgroup of the symmetric group, which we define in this paper. We generalize also the exchange laws for the alternating subgroup, and we will show some interesting properties of it.