Abstract: In this paper, we aimed at determining all subgroups of the Symmetric group S5 up to Automorphism class using Sylow's theorem and Lagrange's theorem. This is achieved by finding all subgroups of order m for which m|O(S5) and are subsets of S5. It was vividly described and derived 156 subgroups of S5 and their conjugacy class size and Isomorphism class. The Alternating group A5 is the unique maximal normal subgroup of S5. Further, the Symmetric group S5 is centerless and every automorphism of it is inner. Also, every natural homomorphism to the automorphism group is an isomorphism. Hence, S5 is complete. The derived subgroups can be used to determine the number of Fuzzy subgroups of the symmetric group S5 for further research.
Keywords: Symmetric group, Conjugacy class, Isomorphism, Automorphism, Complete group
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