Groups with elements of order 8 do not have the DCI property
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Date
2025
Authors
Dobson, Ted
Morris, Joy
Spiga, Pablo
Journal Title
Journal ISSN
Volume Title
Publisher
University of Primorska
Abstract
Let k be odd, and n an odd multiple of 3. Although this can also be deduced from known results, we provide a new proof that Ck ⋊ C₈ and (Cn × C₃) ⋊ C₈ do not have the Directed Cayley Isomorphism (DCI) property. When k is prime, Ck ⋊ C₈ had previously been proved to have the Cayley Isomorphism (CI) property. To the best of our knowledge, the groups Cp ⋊ 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 provides a new proof of the result (which follows from known results but was not explicitly published) that no group with an element of order 8 has the DCI property.
One piece of our proof is a new result that may prove to be of independent interest: we show that if a permutation group has a regular
subgroup of index 2 then it must be 2-closed.
Description
Open access article. Creative Commons Attribution license (CC BY 4.0) applies
Keywords
CI property , DCI property , Cayley graphs , Cayley digraphs , 2-closed groups
Citation
Dobson, T., Morris, J., & Spiga, P. (2025). Groups with elements of order 8 do not have the DCI property. The Art of Discrete and Applied Mathematics, 8(2), Article #P2.08. https://doi.org/10.26493/2590-9770.1549.3cb