On supersolvability of fatorized finite groups
Tóm tắt
In this paper, we investigate the structure of finite groups that are products of two supersolvable groups and gain a sufficient condition for a group to be supersolvable. Our main theorem is the following: Let the group
$$G=HK$$
be the product of the subgroups
$$H$$
and
$$K$$
. Assume that
$$H$$
permutes with every maximal subgroup of
$$K$$
and
$$K$$
permutes with every maximal subgroup of
$$H$$
. If
$$H$$
is supersolvable, and
$$K$$
is nilpotent and
$$K$$
is
$$\delta $$
-permutable in
$$H$$
, where
$$\delta $$
is a complete set of Sylow subgroups of
$$H$$
, then
$$G$$
is supersolvable. Some known results are generalized.
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