Structure of $k$-closures of finite nilpotent permutation groups

Abstract: Let

G

be a permutation group on a set

O m e g a

, and

k

a positive integer. The

k

-closure

G^{(k)}

of

G

is the largest subgroup of

o p e r a t o r n a m e S y m (O m e g a)

, with the same as

G

orbits of componentwise action on

O m e g a^{k}

. We prove that the

k

-closure of a finite nilpotent permutation group is the direct product of

k

-closures of its Sylow subgroups.

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