Finite groups with specific numbers of cyclic subgroups satisfy solvability or supersolvability, with a partial extension of the classification of n-cyclic groups for n at least 13.
Group Structure via Subgroup Counts
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abstract
The number of subgroups and the number of cyclic subgroups are natural combinatorial invariants of a finite group. We investigate how restrictions on these quantities, together with the number of distinct prime divisors of $|G|$, enforce nilpotency, supersolvability, and solvability of $G$. These criteria improve earlier results that relied solely on the total number of subgroups, and they are sharp in the sense that for each bound there exist non-nilpotent (respectively non-supersolvable, non-solvable) groups attaining the bound.
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math.GR 1years
2026 1verdicts
UNVERDICTED 1representative citing papers
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Solvability of Groups via Cyclic Subgroup Count
Finite groups with specific numbers of cyclic subgroups satisfy solvability or supersolvability, with a partial extension of the classification of n-cyclic groups for n at least 13.