| Title:
|
A new characterization of symmetric group by NSE (English) |
| Author:
|
Babai, Azam |
| Author:
|
Akhlaghi, Zeinab |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
67 |
| Issue:
|
2 |
| Year:
|
2017 |
| Pages:
|
427-437 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $G$ be a group and $\omega (G)$ be the set of element orders of $G$. Let $k\in \omega (G)$ and $m_k(G)$ be the number of elements of order $k$ in $G$. Let nse$(G) = \{m_k(G) \colon k \in \omega (G)\}$. Assume $r$ is a prime number and let $G$ be a group such that nse$(G)=$ nse$(S_r)$, where $S_r$ is the symmetric group of degree $r$. In this paper we prove that $G\cong S_r$, if $r$ divides the order of $G$ and $r^2$ does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components. (English) |
| Keyword:
|
set of the numbers of elements of the same order |
| Keyword:
|
prime graph |
| MSC:
|
20D06 |
| MSC:
|
20D15 |
| idZBL:
|
Zbl 06738529 |
| idMR:
|
MR3661051 |
| DOI:
|
10.21136/CMJ.2017.0700-15 |
| . |
| Date available:
|
2017-06-01T14:29:17Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146766 |
| . |
| Reference:
|
[1] Ahanjideh, N., Asadian, B.: NSE characterization of some alternating groups.J. Algebra Appl. 14 (2015), Article ID 1550012, 14 pages. Zbl 1320.20016, MR 3270051, 10.1142/S0219498815500127 |
| Reference:
|
[2] Asboei, A. K.: A new characterization of PGL$(2,p)$.J. Algebra Appl. 12 (2013), Article ID 1350040, 5 pages. Zbl 1278.20013, MR 3063479, 10.1142/S0219498813500400 |
| Reference:
|
[3] Asboei, A. K., Amiri, S. S. S., Iranmanesh, A., Tehranian, A.: A characterization of symmetric group $S_r$, where $r$ is prime number.Ann. Math. Inform. 40 (2012), 13-23. Zbl 1261.20025, MR 3005112 |
| Reference:
|
[4] Frobenius, G.: Verallgemeinerung des Sylow'schen Satzes.Berl. Ber. (1895), 981-993 German \99999JFM99999 26.0158.01. 10.3931/e-rara-18880 |
| Reference:
|
[5] Gorenstein, D.: Finite Groups.Harper's Series in Modern Mathematics, Harper and Row, Publishers, New York (1968). Zbl 0185.05701, MR 0231903 |
| Reference:
|
[6] Gruenberg, K. W., Roggenkamp, K. W.: Decomposition of the augmentation ideal and of the relation modules of a finite group.Proc. Lond. Math. Soc., III. Ser. 31 (1975), 149-166. Zbl 0313.20004, MR 0374247, 10.1112/plms/s3-31.2.149 |
| Reference:
|
[7] M. Hall, Jr.: The Theory of Groups.The Macmillan Company, New York (1959). Zbl 0084.02202, MR 0103215 |
| Reference:
|
[8] Huppert, B.: Endliche Gruppen. I.Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen 134, Springer, Berlin German (1967). Zbl 0217.07201, MR 0224703, 10.1007/978-3-642-64981-3 |
| Reference:
|
[9] Khatami, M., Khosravi, B., Akhlaghi, Z.: A new characterization for some linear groups.Monatsh. Math. 163 (2011), 39-50. Zbl 1216.20022, MR 2787581, 10.1007/s00605-009-0168-1 |
| Reference:
|
[10] Kondrat'ev, A. S., Mazurov, V. D.: Recognition of alternating groups of prime degree from their element orders.Sib. Math. J. 41 (2000), 294-302 translation from Sib. Mat. Zh. 41 359-369 Russian 2000. Zbl 0956.20007, MR 1762188, 10.1007/BF02674599 |
| Reference:
|
[11] Shao, C., Jiang, Q.: A new characterization of some linear groups by nse.J. Algebra Appl. 13 (2014), Article ID 1350094, 9 pages. Zbl 1286.20021, MR 3119655, 10.1142/S0219498813500941 |
| Reference:
|
[12] Shi, W. J.: A new characterization of the sporadic simple groups.Group Theory Proc. Conf., Singapore, 1987, Walter de Gruyter, Berlin (1989), 531-540. Zbl 0657.20017, MR 0981868, 10.1515/9783110848397-040 |
| Reference:
|
[13] Weisner, L.: On the Sylow subgroups of the symmetric and alternating groups.Am. J. Math. 47 (1925), 121-124 \99999JFM99999 51.0117.02. MR 1506549, 10.2307/2370639 |
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