×

On the number of maximal theta pairs in a finite group. (English) Zbl 1028.20017

Summary: N. P. Mukherjee and P. Bhattacharya [Proc. Am. Math. Soc. 109, No. 3, 589-596 (1990; Zbl 0699.20019)] defined the notion of \(\theta\)-pair for a maximal subgroup of a finite group. They proved that for any maximal subgroup \(M\) of a finite group \(G\), there exists a \(\theta\)-pair related to \(M\). Y.-Q. Zhao [Commun. Algebra 23, No. 6, 2099-2106 (1995; Zbl 0830.20045)] improved this result. He proved that for any maximal subgroup \(M\) of a finite group \(G\), there exists a normal maximal \(\theta\)-pair related to \(M\).
In this paper we introduce the notions of \(n\theta\)-maximal and primitive \(n\theta\)-maximal group. We show that for \(n=1,2\), \(G\) is \(n\theta\)-maximal if and only if \(G\) is primitive \(n\theta\)-maximal. Also, we characterize the \(1\theta\)-maximal groups and prove some results about \(2\theta\)-maximal groups. Finally, we introduce the notion of \(n\theta\)-pair group and prove that for all \(n\neq 2,3\), there exist \(n\theta\)-pair groups and for \(n=2,3\) there is no \(n\theta\)-pair group.

MSC:

20D25 Special subgroups (Frattini, Fitting, etc.)
20E28 Maximal subgroups
PDF BibTeX XML Cite
Full Text: EuDML

References:

[1] A. Ballester-Bolinches, Zhao Yaoqing: On maximal subgroups of finite groups and theta pairs. Comm. in Algebra, 24 (13) (1996), 4199-4209. · Zbl 0892.20021
[2] J. C. Beidlemen, H. Smith: A note on supersolvable maximal subgroups and theta pairs. Publicacious Matematiques, 37 (1993), 91-94. · Zbl 0803.20021
[3] D. Gorenstein: Finite Groups. New York, 1968. · Zbl 0185.05701
[4] B. Huppert: Endliche Gruppen. Springer-Verlag, Berlin, 1967. · Zbl 0217.07201
[5] R. R. Khazal: A note on minimal and maximal subgroups in a finite group. J. Univ. Kuwait (Sci.), 16 (1989), 229-235. · Zbl 0686.20017
[6] N. P. Mukherjee, P. Bhattacharya: On theta pairs for a maximal subgroup of a finite group. Proc. Amer. Math. Soc, 109 (1990), 589-596. · Zbl 0699.20019
[7] Li Shirong: A Note on Theta Pairs for Maximal Subgroups. Comm. in Algebra, 26 (12) (1998), 4277-4284. · Zbl 0919.20013
[8] R. Soleimani: On the Number of Maximal Theta Pairs in a Finite Groups. M.Sc. Thesis, University of Kashan, 2000.
[9] M. Weinstein: Between nilpotent and solvable. Polygonal Publishing House, Washington, NJ., 1981. · Zbl 0488.20001
[10] Guo Xiuyun: On theta pairs for a maximal subgroup. Comm. in Algebra, 22 (12) (1994), 4653-4659. · Zbl 0814.20016
[11] Zhao Yaoqing: On theta pairs for maximal subgroups. Comm. in Algebra, 23 (6) (1995), 2099-2106. · Zbl 0830.20045
[12] Zhao Yaoqing: On the Deskins completions, theta pairs and theta completions for maximal subgroups. Far East J. Math. Sci. (FJMS), 1(5) (1999), 827-835. · Zbl 0997.20026
[13] Zhao Yaoqing: On the Deskins completions, theta completions and theta pairs for maximal subgroups. Comm. in Algebra, 26 (10) (1998), 3141-3153. · Zbl 0911.20018
[14] Zhao Yaoqing: On the Deskins completions, theta completions and theta pairs for maximal subgroups II. Comm. in Algebra, 26 (10) (1998), 3155-3164. · Zbl 0911.20018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.