Finite Abelian Groups with Isomorphic Inclusion Graphs of Cyclic Subgroups

Document Type : Original paper

Authors

Faculty of Mathematical Science, Shahrood University of Technology, Shahrood, I.R. Iran

Abstract

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.

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References

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Communications in Combinatorics and Optimization
Volume 10, Issue 1
March 2025
Pages 165-180

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