Any finite group with the same order and conjugacy class size set as an alternating or symmetric group is isomorphic to it.
Lusztig, On the group attached to a special Weyl group representation, arxiv:2505.00230 (2025)
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Characterization of the alternating and symmetric groups by the order and conjugacy class sizes
Any finite group with the same order and conjugacy class size set as an alternating or symmetric group is isomorphic to it.