Groups with elements of order 8 do not have the DCI property

TL;DR

This paper demonstrates that certain groups with elements of order 8 lack the Directed Cayley Isomorphism (DCI) property, highlighting a distinction between CI and DCI properties in group theory.

Contribution

It proves that specific groups with elements of order 8 do not possess the DCI property, providing new examples and insights into the relationship between CI and DCI properties.

Findings

Groups with elements of order 8 do not have the DCI property.

Certain groups with order 8 elements have the CI property but not the DCI property.

No group with an element of order 8 has the DCI property.

Abstract

Let $k$ be odd, and $n$ an odd multiple of $3$. We prove that $C_k \rtimes C_8$ and $(C_n \times C_3)\rtimes C_8$ do not have the Directed Cayley Isomorphism (DCI) property. When $k$ is also prime, $C_k \rtimes C_8$ had previously been proved to have the Cayley Isomorphism (CI) property. To the best of our knowledge, the groups $C_p \rtimes C_8$ (where $p$ is an odd prime) are only the second known infinite family of groups that have the CI property but do not have the DCI property. This also shows that no group with an element of order $8$ has the DCI property.