$CI$-property for decomposable Schur rings over an abelian group

TL;DR

This paper provides a sufficient condition for decomposable Schur rings over direct products of elementary abelian groups to have the $CI$-property, simplifying proofs of known results for groups of rank up to 5.

Contribution

It introduces a new sufficient condition for the $CI$-property in decomposable Schur rings over certain abelian groups, streamlining existing proofs.

Findings

Established a sufficient condition for $CI$-property

Reproved known results for rank ≤ 5 groups

Simplified proofs of $CI$-property in specific cases

Abstract

A Schur ring over a finite group is said to be decomposable if it is the generalized wreath product of Schur rings over smaller groups. In this paper we establish a sufficient condition for a decomposable Schur ring over the direct product of elementary abelian groups to be a $CI$-Schur ring. By using this condition we reprove in a short way known results on the $CI$-property for decomposable Schur rings over an elementary abelian group of rank at most $5$.