Conjugation groups and structure groups of quandles
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911955419136000 |
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| author | Lebed, Victoria |
| author_facet | Lebed, Victoria |
| contents | Quandles are certain algebraic structures showing up in different mathematical contexts. A group $G$ with the conjugation operation forms a quandle, $\operatorname{Conj}(G)$. In the opposite direction, one can construct a group $\operatorname{As}(Q)$ starting from any quandle $Q$. These groups are useful in practice, but hard to compute. We explore the group $\operatorname{As}(\operatorname{Conj}(G))$ for so-called $\overline{C}$-groups $G$. These are groups admitting a presentation with only conjugation and power relations. Symmetric groups $S_n$ are typical examples. We show that for $\overline{C}$-groups, $\operatorname{As}(\operatorname{Conj}(G))$ injects into $G \times \mathbb{Z}^m$, where $m$ is the number of conjugacy classes of $G$. From this we deduce information about the torsion, center, and derived group of $\operatorname{As}(\operatorname{Conj}(G))$. As an application, we compute the second quandle homology group of $\operatorname{Conj}(S_n)$ for all $n$, and unveil rich torsion therein. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02955 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conjugation groups and structure groups of quandles Lebed, Victoria Group Theory 57K12 (Primary) 20B30, 20F05, 16S15, 16T25 (Secondary) Quandles are certain algebraic structures showing up in different mathematical contexts. A group $G$ with the conjugation operation forms a quandle, $\operatorname{Conj}(G)$. In the opposite direction, one can construct a group $\operatorname{As}(Q)$ starting from any quandle $Q$. These groups are useful in practice, but hard to compute. We explore the group $\operatorname{As}(\operatorname{Conj}(G))$ for so-called $\overline{C}$-groups $G$. These are groups admitting a presentation with only conjugation and power relations. Symmetric groups $S_n$ are typical examples. We show that for $\overline{C}$-groups, $\operatorname{As}(\operatorname{Conj}(G))$ injects into $G \times \mathbb{Z}^m$, where $m$ is the number of conjugacy classes of $G$. From this we deduce information about the torsion, center, and derived group of $\operatorname{As}(\operatorname{Conj}(G))$. As an application, we compute the second quandle homology group of $\operatorname{Conj}(S_n)$ for all $n$, and unveil rich torsion therein. |
| title | Conjugation groups and structure groups of quandles |
| topic | Group Theory 57K12 (Primary) 20B30, 20F05, 16S15, 16T25 (Secondary) |
| url | https://arxiv.org/abs/2407.02955 |