Finite groups with 30 elements of maximal order. (English) Zbl 1148.20015
Let \(G\) be a finite group. Let \(M(G)\) denote the number of elements of maximal order \(k\) in \(G\). In the paper under review the authors prove that if \(M(G)=30\), then \(k\) is equal to one of the values: 6, 9, 18, 22, 31 or \(62\). In each case the authors give restriction on the order of \(G\) and its structure.
Reviewer: Mohammad-Reza Darafsheh (Tehran)
MSC:
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20E34 | General structure theorems for groups |
| 20E45 | Conjugacy classes for groups |
References:
| [1] | Cheng, Y.: Finite groups based on the numbers of elements of maximal order (in Chinese). Chinese Ann. Math. Ser. A 14(5), 561–567 (1993) · Zbl 0829.20036 |
| [2] | Youyi, J.: Finite groups with the number not exceeding 19 of elements of maximal order are solvable (in Chinese). J. Southwest China Normal Univ. (Natural Science Edition) 23(4), 379–384 (1998) |
| [3] | Youyi, J.: Finite groups with 2p 2 elements of maximal order are solvable (in Chinese). Chin. Ann. Math., A 21(1), 61–64 (2000) · Zbl 0964.20008 |
| [4] | He, C.: Finite groups with 2m ( \((m,\;30)=1\) ) elements of maximal order. J. Southwest China Normal Univ. (Natural Science Edition) 29(3), 351–353 (2004) |
| [5] | Williams, J.S.: Prime graph components of finite groups. J. Algebra 69, 487–513 (1981) · Zbl 0471.20013 · doi:10.1016/0021-8693(81)90218-0 |
| [6] | Kondratev, A.S.: Prime graph components of finite simple groups. Math. USSR Sbornik. 67(1), 235–247 (1990) · Zbl 0698.20009 · doi:10.1070/SM1990v067n01ABEH001363 |
| [7] | Chen, G.Y.: On Frobenius groups and 2-Frobenius groups. J. Southwest China Normal Univ. (Natural Science Edition) 20(5), 485–487 (1995) |
| [8] | Chen, G.Y.: A new characterization of sporadic simple groups. Algebra Colloq. 3(1), 49–58 (1996) · Zbl 0845.20011 |
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