## Abstract

For a nontrivial finite group *G*, the intersection graph \(\Gamma (G)\) of *G* is the simple undirected graph whose vertices are the nontrivial proper subgroups of *G* and two vertices are joined by an edge if and only if they have a nontrivial intersection. In a finite simple graph \(\Gamma \), the clique number of \(\Gamma \) is denoted by \(\omega (\Gamma )\). In this paper we show that if *G* is a finite group with \(\omega (\Gamma (G))<13\), then *G* is solvable. As an application, we characterize all non-solvable groups *G* with \(\omega (\Gamma (G))=13\). Moreover, we determine all finite groups *G* with \(\omega (\Gamma (G))\in \{2,3,4\}\).

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Communicated by Mohammad Zarrin.

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Beheshtipour, A., Jafarian Amiri, S.M. The Clique Number of the Intersection Graph of a Finite Group.
*Bull. Iran. Math. Soc.* **49**, 74 (2023). https://doi.org/10.1007/s41980-023-00804-5

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DOI: https://doi.org/10.1007/s41980-023-00804-5