The Sylow Dimension of Finite Groups: A Novel Structural Invariant
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2025
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| _version_ | 1866901198804615168 |
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| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | This paper introduces the concept of the Sylow dimension of a finite group, a novel structural invariant designed to quantify the complexity of a group's Sylow structure. The Sylow dimension, denoted SDim(G), is defined as the number of distinct prime divisors p of the group order $|G|$ for which the Sylow p-subgroup of G is not normal in G. This invariant provides a refined measure beyond traditional classifications, offering new insights into the interplay between local p-structure and global group properties. We establish fundamental properties of the Sylow dimension, demonstrating that a group G is nilpotent if and only if SDim(G) = 0. We explore its behavior under various group operations, including direct products and quotients, and provide characterizations for groups with small Sylow dimensions. Through detailed analysis and illustrative examples, this work demonstrates the utility of Sylow dimension in distinguishing group structures and sheds new light on the intricate relationships between Sylow p-subgroups and the overall architecture of finite groups. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17689801 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Sylow Dimension of Finite Groups: A Novel Structural Invariant SÉRGIO DE ANDRADE, PAULO This paper introduces the concept of the Sylow dimension of a finite group, a novel structural invariant designed to quantify the complexity of a group's Sylow structure. The Sylow dimension, denoted SDim(G), is defined as the number of distinct prime divisors p of the group order $|G|$ for which the Sylow p-subgroup of G is not normal in G. This invariant provides a refined measure beyond traditional classifications, offering new insights into the interplay between local p-structure and global group properties. We establish fundamental properties of the Sylow dimension, demonstrating that a group G is nilpotent if and only if SDim(G) = 0. We explore its behavior under various group operations, including direct products and quotients, and provide characterizations for groups with small Sylow dimensions. Through detailed analysis and illustrative examples, this work demonstrates the utility of Sylow dimension in distinguishing group structures and sheds new light on the intricate relationships between Sylow p-subgroups and the overall architecture of finite groups. |
| title | The Sylow Dimension of Finite Groups: A Novel Structural Invariant |
| url | https://doi.org/10.5281/zenodo.17689801 |