| Title:
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Several quantitative characterizations of some specific groups (English) |
| Author:
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Mohammadzadeh, A. |
| Author:
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Moghaddamfar, A. R. |
| Language:
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English |
| Journal:
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Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
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0010-2628 (print) |
| ISSN:
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1213-7243 (online) |
| Volume:
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58 |
| Issue:
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1 |
| Year:
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2017 |
| Pages:
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19-34 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $G$ be a finite group and let $\pi(G)=\{p_1, p_2,\ldots, p_k\}$ be the set of prime divisors of $|G|$ for which $p_1< p_2< \cdots < p_k$. The Gruenberg-Kegel graph of $G$, denoted $\operatorname{GK} (G)$, is defined as follows: its vertex set is $\pi(G)$ and two different vertices $p_i$ and $p_j$ are adjacent by an edge if and only if $G$ contains an element of order $p_i p_j$. The degree of a vertex $p_i$ in ${\rm GK}(G)$ is denoted by $d_G(p_i)$ and the $k$-tuple $D(G)= (d_G(p_1), d_G(p_2),\ldots, d_G(p_k))$ is said to be the degree pattern of $G$. Moreover, if $\omega \subseteq \pi(G)$ is the vertex set of a connected component of $\operatorname{GK} (G)$, then the largest $\omega$-number which divides $|G|$, is said to be an order component of $\operatorname{GK} (G)$. We will say that the problem of OD-characterization is solved for a finite group if we find the number of pairwise non-isomorphic finite groups with the same order and degree pattern as the group under study. The purpose of this article is twofold. First, we completely solve the problem of OD-characterization for every finite non-abelian simple group with orders having prime divisors at most 29. In particular, we show that there are exactly two non-isomorphic finite groups with the same order and degree pattern as $U_4(2)$. Second, we prove that there are exactly two non-isomorphic finite groups with the same order components as $U_5(2)$. (English) |
| Keyword:
|
OD-characterization of finite group |
| Keyword:
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prime graph |
| Keyword:
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degree pattern |
| Keyword:
|
simple group |
| Keyword:
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$2$-Frobenius group |
| MSC:
|
20D05 |
| MSC:
|
20D06 |
| MSC:
|
20D08 |
| idZBL:
|
Zbl 06736741 |
| idMR:
|
MR3631678 |
| DOI:
|
10.14712/1213-7243.2015.194 |
| . |
| Date available:
|
2017-03-12T16:35:11Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146025 |
| . |
| Reference:
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