Orbit isomorphic skeleton groups

Recent development in the classification of \(p\)-groups often concentrate on the coclass graph \(\mathcal{G}(p,r)\) associated with the finite \(p\)-groups coclass \(r\), specially on periodicity results on these graphs. In particular, the structure of the subgraph induced by `skeleton groups'...

Повний опис

Збережено в:
Бібліографічні деталі
Дата:2023
Автор: Saha, S.
Формат: Стаття
Мова:English
Опубліковано: Lugansk National Taras Shevchenko University 2023
Теми:
Онлайн доступ:https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1886
Теги: Додати тег
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Назва журналу:Algebra and Discrete Mathematics

Репозитарії

Algebra and Discrete Mathematics
Опис
Резюме:Recent development in the classification of \(p\)-groups often concentrate on the coclass graph \(\mathcal{G}(p,r)\) associated with the finite \(p\)-groups coclass \(r\), specially on periodicity results on these graphs. In particular, the structure of the subgraph induced by `skeleton groups' is of notable interest. Given their importance, in this paper, we investigate periodicity results of skeleton groups. Our results concentrate on the skeleton groups in \(\mathcal{G}(p,1)\). We find a family of skeleton groups in \(\mathcal{G}(7,1)\) whose \(6\)-step parent is not a periodic parent. This shows that the periodicity results available in the current literature for primes \(p\equiv 5\bmod 6\) do not hold for the primes \(p\equiv 1\bmod 6\). We also improve a known periodicity result in a special case of skeleton groups.