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

2024/04/22 by Ted Dobson, Dobson, Ted, Joy Morris +3
Engineering · Mathematics · #05C25 #05E18 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2404.13938

openalex publication_date 2024/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let k be odd, and n an odd multiple of 3. We prove that C_k \rtimes C₈ and (C_n × C₃)\rtimes C₈ do not have the Directed Cayley Isomorphism (DCI) property. When k is also prime, C_k \rtimes C₈ had previously been proved to have the Cayley Isomorphism (CI) property. To the best of our knowledge, the groups C_p \rtimes C₈ (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.

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