Orders of products of horizontal class transpositions
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916402849382400 |
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| author | Bardakov, V. G. Iskra, A. L. |
| author_facet | Bardakov, V. G. Iskra, A. L. |
| contents | The class transposition group $CT(\mathbb{Z})$ was introduced by S. Kohl in 2010. It is a countable subgroup of the permutation group $Sym(\mathbb{Z})$ of the set of integers $\mathbb{Z}$.
We study products of two class transpositions $CT(\mathbb{Z})$ and give a partial answer to the question 18.48 posed by S. Kohl in the Kourovka notebook. We prove that in the group $CT_{\infty}$, which is a subgroup of $CT(\mathbb{Z})$ and generated by horizontal class transpositions, the order of the product of a pair of horizontal class transpositions belongs to the set $\{1,2,3,4,6,12\}$, and any number from this set is the order of the product of a pair of horizontal class transpositions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13341 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Orders of products of horizontal class transpositions Bardakov, V. G. Iskra, A. L. Group Theory 20E07, 20F36, 57K12 The class transposition group $CT(\mathbb{Z})$ was introduced by S. Kohl in 2010. It is a countable subgroup of the permutation group $Sym(\mathbb{Z})$ of the set of integers $\mathbb{Z}$. We study products of two class transpositions $CT(\mathbb{Z})$ and give a partial answer to the question 18.48 posed by S. Kohl in the Kourovka notebook. We prove that in the group $CT_{\infty}$, which is a subgroup of $CT(\mathbb{Z})$ and generated by horizontal class transpositions, the order of the product of a pair of horizontal class transpositions belongs to the set $\{1,2,3,4,6,12\}$, and any number from this set is the order of the product of a pair of horizontal class transpositions. |
| title | Orders of products of horizontal class transpositions |
| topic | Group Theory 20E07, 20F36, 57K12 |
| url | https://arxiv.org/abs/2409.13341 |