On groups with few subgroups not in the Chermak-Delgado lattice
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910549334294528 |
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| author | Burrell, David Cocke, William McCulloch, Ryan |
| author_facet | Burrell, David Cocke, William McCulloch, Ryan |
| contents | We investigate the question of how many subgroups of a finite group are not in its Chermak-Delgado lattice. The Chermak-Delgado lattice for a finite group is a self-dual lattice of subgroups with many intriguing properties. Fasolă and Tărnăuceanu asked how many subgroups are not in the Chermak-Delgado lattice and classified all groups with two or less subgroups not in the Chermak-Delgado lattice. We extend their work by classifying all groups with less than five subgroups not in the Chermak-Delgado lattice. In addition, we show that a group with less than five subgroups not in the Chermak--Delgado lattice is nilpotent. In this vein we also show that the only non-nilpotent group with five or fewer subgroups in the Chermak-Delgado lattice is S_3. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_14910 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On groups with few subgroups not in the Chermak-Delgado lattice Burrell, David Cocke, William McCulloch, Ryan Group Theory We investigate the question of how many subgroups of a finite group are not in its Chermak-Delgado lattice. The Chermak-Delgado lattice for a finite group is a self-dual lattice of subgroups with many intriguing properties. Fasolă and Tărnăuceanu asked how many subgroups are not in the Chermak-Delgado lattice and classified all groups with two or less subgroups not in the Chermak-Delgado lattice. We extend their work by classifying all groups with less than five subgroups not in the Chermak-Delgado lattice. In addition, we show that a group with less than five subgroups not in the Chermak--Delgado lattice is nilpotent. In this vein we also show that the only non-nilpotent group with five or fewer subgroups in the Chermak-Delgado lattice is S_3. |
| title | On groups with few subgroups not in the Chermak-Delgado lattice |
| topic | Group Theory |
| url | https://arxiv.org/abs/2211.14910 |