Elementary Abelian p-groups of rank 2p+3 are not CI-groups

TL;DR

The paper demonstrates that elementary Abelian p-groups of rank at least 2p+3 are not CI-groups by constructing specific Cayley graphs, using elementary algebraic methods, and explains recent bounds uniformly.

Contribution

It provides the first explicit construction of non-CI Cayley graphs for these groups and introduces an elementary algebraic proof technique.

Findings

Elementary Abelian p-groups of rank ≥ 2p+3 are not CI-groups.

Constructs explicit non-CI Cayley graphs for these groups.

Offers a uniform algebraic explanation for existing bounds.

Abstract

For every prime $p > 2$ we exhibit a Cayley graph of $\mathbb{Z}_p^{2p+3}$ which is not a CI-graph. This proves that an elementary Abelian $p$-group of rank greater than or equal to $2p+3$ is not a CI-group. The proof is elementary and uses only multivariate polynomials and basic tools of linear algebra. Moreover, we apply our technique to give a uniform explanation for the recent works concerning the bound.