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Finite Abelian Groups with Isomorphic Inclusion Graphs of Cyclic Subgroups | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 10، دوره 10، شماره 1، خرداد 2025، صفحه 165-180 اصل مقاله (437.07 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2023.27806.1356 | ||
| نویسندگان | ||
| Zahra Gharibbolooki؛ Sayyed Heidar Jafari* | ||
| Faculty of Mathematical Science, Shahrood University of Technology, Shahrood, I.R. Iran | ||
| چکیده | ||
| Let $G$ be a finite group. The directed inclusion graph of cyclic subgroups of $G$, $\overrightarrow{\mathcal{I}_c}(G)$, is the digraph with vertices of all cyclic subgroups of $G$, and for two distinct cyclic subgroups $\langle a \rangle$ and $\langle b \rangle$, there is an arc from $\langle a\rangle $ to $\langle b\rangle $ if and only if $\langle b\rangle \subset \langle a\rangle $. The (undirected ) inclusion graph of cyclic subgroups of $G$, $\mathcal{I}_c(G)$, is the underlying graph of $\overrightarrow{\mathcal{I}_c}(G)$, that is, the vertex set is the set of all cyclic subgroups of $G$ and two distinct cyclic subgroups $\langle a \rangle$ and $\langle b \rangle$ are adjacent if and only if $\langle a\rangle \subset \langle b\rangle$ or $\langle b\rangle \subset \langle a\rangle $. In this paper, we first show that, if $G$ and $H$ are finite groups such that $\mathcal{I}_c(G)\cong \mathcal{I}_c(H)$ and $G$ is cyclic, then $H$ is cyclic. We show that for two cyclic groups $G$ and $H$ of orders $p_1^{\alpha_1} \dots p_t^{\alpha_t}$ and $q_1^{\beta_1} \dots q_s^{\beta_s}$, respectively, $\mathcal{I}_c(G)\cong \mathcal{I}_c(H)$ if and only if $t=s$ and by a suitable $\sigma $, $\alpha_i=\beta_{\sigma (i)}$. Also for any cyclic groups $G,~H$, if $\mathcal{I}_c(G)\cong \mathcal{I}_c(H)$, then $\overrightarrow{\mathcal{I}_c}(G) \cong \overrightarrow{\mathcal{I}_c}(H)$. We also show that for two finite abelian groups $G$ and $H$, $\mathcal{I}_c(G)\cong \mathcal{I}_c(H)$ if and only if $|\pi (G)|=|\pi (H)|$ and by a convenient permutation the graph of their sylow subgroups are isomorphic. In this case, their directed inclusion graphs are isomorphic too. | ||
| کلیدواژهها | ||
| inclusion graph؛ power graph؛ cyclic subgroup؛ abelian group | ||
| مراجع | ||
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