The 2-closure
G
¯
of a permutation group
G
on
Ω
is defined to be the largest permutation group on
Ω
, having the same orbits on
Ω
×
Ω
as
G
. It is proved that if
G
is supersolvable, then
G
¯
can be ...found in polynomial time in
|
Ω
|
. As a by-product of our technique, it is shown that the composition factors of
G
¯
are cyclic or alternating.
The 2-closure
of a permutation group
on a finite set
is the largest subgroup of Sym(
) which has the same orbits as
in the induced action on
×
. In this paper, the 2-closures of certain primitive ...permutation groups of holomorph simple and holomorph compound types are determined.
In this paper we are concerned with 3-groups. We prove that an elementary abelian 3-group of rank 5 is a
CI
(
2
)
-group, and that an elementary abelian 3-group of rank greater than or equal to 8 is ...not a
CI
-group. In Section 4, we present a conjecture.
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