| Title: | Groups with elements of order 8 do not have the DCI property |
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| Authors: | ID Dobson, Ted (Author)
ID Morris, Joy (Author)
ID Spiga, Pablo (Author) |
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| Files: |  ADAM_Dobson,Morris,Spiga_2025.pdf (344,18 KB)
MD5: 1C46AD5A0034E1646033C0EC6A41A2C9
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| Language: | English |
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| Work type: | Article |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: | ZUP - University of Primorska Press
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| 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. |
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| Keywords: | CI property, DCI property, Cayley graphs, Cayley digraphs, 2-closed groups, 2-closure |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 25.03.2025 |
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| Publisher: | Založba Univerze na Primorskem |
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| Year of publishing: | 2025 |
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| Number of pages: | 5 str. |
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| Numbering: | Vol. 8, no. 2, [article no.] P2.08 |
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| PID: | 20.500.12556/RUP-22071  |
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| UDC: | 51 |
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| eISSN: | 2590-9770 |
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| DOI: | 10.26493/2590-9770.1549.3cb |
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| Publication date in RUP: | 03.11.2025 |
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| Views: | 169 |
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| Downloads: | 1 |
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