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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.07327 |
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| _version_ | 1866912320349798400 |
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| author | Giorgi, Alessandro |
| author_facet | Giorgi, Alessandro |
| contents | We study the finite solvable groups $G$ in which every real element has prime power order. We divide our examination into two parts: the case $\textbf{O}_2(G)>1$ and the case $\textbf{O}_2(G)=1$. Specifically we proved that if $\textbf{O}_2(G)>1$ then $G$ is a $\{2,p\}$-group. Finally, by taking into consideration the examples presented in the analysis of the $\textbf{O}_2(G)=1$ case, we deduce some interesting and unexpected results about the connectedness of the real prime graph $Γ_{\mathbb{R}}(G)$. In particular, we found that there are groups such that $Γ_{\mathbb{R}}(G)$ has respectively 3 and 4 connected components. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_07327 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solvable Groups in which Every Real Element has Prime Power Order Giorgi, Alessandro Group Theory We study the finite solvable groups $G$ in which every real element has prime power order. We divide our examination into two parts: the case $\textbf{O}_2(G)>1$ and the case $\textbf{O}_2(G)=1$. Specifically we proved that if $\textbf{O}_2(G)>1$ then $G$ is a $\{2,p\}$-group. Finally, by taking into consideration the examples presented in the analysis of the $\textbf{O}_2(G)=1$ case, we deduce some interesting and unexpected results about the connectedness of the real prime graph $Γ_{\mathbb{R}}(G)$. In particular, we found that there are groups such that $Γ_{\mathbb{R}}(G)$ has respectively 3 and 4 connected components. |
| title | Solvable Groups in which Every Real Element has Prime Power Order |
| topic | Group Theory |
| url | https://arxiv.org/abs/2504.07327 |