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.

**Keywords**

**Main Subjects**

[1] J. Abawajy, A. Kelarev, and M. Chowdhury, Power graphs: A survey, Electron. J. Graph Theory Appl. 1 (2013), no. 2, 125–147. https://dx.doi.org/10.5614/ejgta.2013.1.2.6

[2] P.J. Cameron, The power graph of a finite group, II, J. Group Theory 13 (2010), no. 779–783. https://doi.org/10.1515/jgt.2010.023

[3] P.J. Cameron and S. Ghosh, The power graph of a finite group, Discrete Math. 311 (2011), no. 13, 1220–1222. https://doi.org/10.1016/j.disc.2010.02.011

[4] P.J. Cameron, H. Guerra, and Š. Jurina, The power graph of a torsion-free group, J. Algebraic Combin. 49 (2019), no. 1, 83–98. https://doi.org/10.1007/s10801-018-0819-1

[5] P.J. Cameron and S.H. Jafari, On the connectivity and independence number of power graphs of groups, Graphs Combin. 36 (2020), no. 3, 895–904. https://doi.org/10.1007/s00373-020-02162-z

[6] K.N. Cheng, M. Deaconescu, M.L. Lang, and W.J. Shi, Corrigendum and addendum to: “classification of finite groups with all elements of prime order”, Proc. Amer. Math. Soc. 117 (1993), no. 4, 1205–1207. https://doi.org/10.2307/2159554

[7] M. Deaconescu, Classification of finite groups with all elements of prime order, Proc. Amer. Math. Soc. 106 (1989), no. 3, 625–629. https://doi.org/10.2307/2047414

[8] Z. Garibbolooki and S.H. Jafari, Planarity of inclusion graph of cyclic subgroups of finite group, Math. Interdisc. Res. 5 (2020), no. 4, 303–314. https://doi.org/10.22052/mir.2020.209251.1183

[9] S.H. Jafari and S. Chattopadhyay, Spectrum of proper power graphs of the direct product of certain finite groups, Linear Multilinear Algebra 70 (2022), no. 20, 5460–5481. https://doi.org/10.1080/03081087.2021.1918051

[10] A.V. Kelarev and S.J. Quinn, A combinatorial property and power graphs of groups, Contrib. General Algebra 12 (2000), 229–235.

[11] X. Ma, Proper connection of power graphs of finite groups, J. Algebra . Appl. 20 (2021), no. 3, Article ID: 2150033. https://doi.org/10.1142/S021949882150033X

[12] X. Ma and L. Zhai, Strong metric dimensions for power graphs of finite groups, Comm. Algebra 49 (2021), no. 11, 4577–4587. https://doi.org/10.1080/00927872.2021.1924764

[13] G.R. Pourgholi, H. Yousefi-Azari, and A.R. Ashrafi, The undirected power graph of a finite group, Bull. Malays. Math. Sci. 38 (2015), no. 4, 1517–1525. https://doi.org/10.1007/s40840-015-0114-4

[14] R. Rajkumar and T. Anitha, Reduced power graph of a group, Electron. Notes Discrete Math. 63 (2017), 69–76. https://doi.org/10.1016/j.endm.2017.10.063

[15] R. Rajkumar and T. Anitha, Some results on the reduced power graph of a group, Southeast Asian Bull. Math. 45 (2021), 241–262.

[16] D.J.S. Robinson, A Course in the Theory of Groups, vol. 80, Springer Science & Business Media, 2012.

[17] M. Shaker and M.A. Iranmanesh, On groups with specified quotient power graphs, Int. J. Group Theory 5 (2016), no. 3, 49–60. https://doi.org/10.22108/ijgt.2016.8542

March 2025

Pages 165-180