A note about solvable and non-solvable finite groups of the same order type
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913468559392768 |
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| author | Müller, Peter |
| author_facet | Müller, Peter |
| contents | Two finite groups are said to have the same order type if for each positive integer $n$ both groups have the same number of elements of order $n$. In 1987 John G. Thompson asked if in this case the solvability of one group implies the solvability of the other group.
In 2024 Pawel Piwek gave a negative example. He constructed two groups of order $2^{365}\cdot3^{105}\cdot7^{104}\approx7.3\cdot10^{247}$ of the same order type, where only one is solvable.
In this note we produce a much smaller example of order $2^{13}\cdot3^4\cdot7^3=227598336$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_07732 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note about solvable and non-solvable finite groups of the same order type Müller, Peter Group Theory 20D60 (Primary) Two finite groups are said to have the same order type if for each positive integer $n$ both groups have the same number of elements of order $n$. In 1987 John G. Thompson asked if in this case the solvability of one group implies the solvability of the other group. In 2024 Pawel Piwek gave a negative example. He constructed two groups of order $2^{365}\cdot3^{105}\cdot7^{104}\approx7.3\cdot10^{247}$ of the same order type, where only one is solvable. In this note we produce a much smaller example of order $2^{13}\cdot3^4\cdot7^3=227598336$. |
| title | A note about solvable and non-solvable finite groups of the same order type |
| topic | Group Theory 20D60 (Primary) |
| url | https://arxiv.org/abs/2408.07732 |