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In [6], Bhattacharya and Mukherjee 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 $M$. In [11], Zhao
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 notion 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
group and prove some results about $2\theta$-maximal groups.
Finally, we introduce the notion of $n\theta$-pair group and prove
that for all $n\neq2,3$, there exists $n\theta$-pair groups and
for $n=2,3$ there is no $n\theta$-pair groups.
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