Group Structure via Subgroup Counts
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910115947347968 |
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| author | Das, Angsuman Dey, Hiranya Kishore Sharma, Khyati |
| author_facet | Das, Angsuman Dey, Hiranya Kishore Sharma, Khyati |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_08040 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Group Structure via Subgroup Counts Das, Angsuman Dey, Hiranya Kishore Sharma, Khyati Group Theory Combinatorics 20D15, 20D60, 20E07, 05E16 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. |
| title | Group Structure via Subgroup Counts |
| topic | Group Theory Combinatorics 20D15, 20D60, 20E07, 05E16 |
| url | https://arxiv.org/abs/2604.08040 |