Article
| Title: | On the recognizability of some projective general linear groups by the prime graph (English) |
| Author: | Sajjadi, Masoumeh |
| Language: | English |
| Journal: | Commentationes Mathematicae Universitatis Carolinae |
| ISSN: | 0010-2628 (print) |
| ISSN: | 1213-7243 (online) |
| Volume: | 63 |
| Issue: | 4 |
| Year: | 2022 |
| Pages: | 443-458 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $G $ be a finite group. The prime graph of $G$ is a simple graph $\Gamma(G)$ whose vertex set is $\pi(G)$ and two distinct vertices $p$ and $q$ are joined by an edge if and only if $G$ has an element of order $pq$. A group $ G $ is called $ k $-recognizable by prime graph if there exist exactly $ k$ nonisomorphic groups $ H$ satisfying the condition $ \Gamma(G) = \Gamma(H)$. A 1-recognizable group is usually called a recognizable group. In this problem, it was proved that ${\rm PGL}(2,p^\alpha) $ is recognizable, if $ p$ is an odd prime and $ \alpha > 1$ is odd. But for even $ \alpha $, only the recognizability of the groups $ {\rm PGL}(2, 5^2)$, $ {\rm PGL}(2, 3^2) $ and $ {\rm PGL}(2, 3^4) $ was investigated. In this paper, we put $ \alpha = 2$ and we classify the finite groups $G$ that have the same prime graph as $\Gamma({\rm PGL}(2, p^2))$ for $p=7, 11, 13$ and 17. As a result, we show that ${\rm PGL}(2, 7^2)$ is unrecognizable; and ${\rm PGL}(2, 13^2)$ and ${\rm PGL}(2, 17^2)$ are recognizable by prime graph. (English) |
| Keyword: | projective general linear group |
| Keyword: | prime graph |
| Keyword: | recognition |
| MSC: | 20D05 |
| MSC: | 20D08 |
| MSC: | 20D60 |
| idZBL: | Zbl 07729553 |
| idMR: | MR4577040 |
| DOI: | 10.14712/1213-7243.2023.009 |
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| Date available: | 2023-04-20T13:50:09Z |
| Last updated: | 2023-10-27 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151645 |
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