This study investigates the power graphs of the symmetric groups ____ for 3 _ __ _ 6. In a power graph, vertices represent group elements, and edges connect vertices if one element is a power of the other. The study identifies key structural patterns, including over 250 distinct subgraphs for ____. Specifically, complete subgraphs ____, correspond to cyclic subgroups of prime power orders. In contrast, incomplete subgraphs highlight the non-cyclic nature of symmetric groups for __ > 4, as they cannot be generated by a single element. This research provides a systematic characterization of power graphs for small symmetric groups, offering new insights into their combinatorial structure. Additionally, it contributes to the existing literature by demonstrating how the complexity of power graphs increases with __, revealing intricate power relations and the presence of both cyclic and non-cyclic structural components. Keywords Power graph of group, symmetric group, cyclic group, non-cycli.....

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