## 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

### Keywords:

finite groups; maximal subgroups; $$\theta$$-pairs

### Citations:

Zbl 0699.20019; Zbl 0830.20045
Full Text:

### References:

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