Bounds on the number of subgroups and cyclic subgroups together with the number of distinct prime divisors of |G| imply nilpotency, supersolvability or solvability for finite groups, with sharpness shown by explicit examples.
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Group Structure via Subgroup Counts
Bounds on the number of subgroups and cyclic subgroups together with the number of distinct prime divisors of |G| imply nilpotency, supersolvability or solvability for finite groups, with sharpness shown by explicit examples.