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

Abstract

Let $k$ be odd, and $n$ an odd multiple of $3$. We prove that $C_k⋊C_8$ and $(C_n\times C_3)⋊C_8$ do not have the Directed Cayley Isomorphism (DCI) property. When $k$ is also prime, $C_k⋊C_8$ had previously been proved to have the Cayley Isomorphism (CI) property. To the best of our knowledge, the groups $C_p⋊C_8$ (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.

4 pages