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On the 2-closures of finite permutation groups. (English) Zbl 0655.20003
The concept of the k-closure $$G^{(k)}$$ of a permutation group G acting on $$\Omega$$ was introduced by H. Wielandt [Permutation groups through invariant relations and invariant functions (Lecture Notes, Ohio State Univ. 1969)]: $$G^{(k)}$$ is by definition the largest subgroup of Sym($$\Omega)$$ containing G and having the same orbits as G in the action induced on $$\Omega$$ k.
In the present paper the important special case is considered where G is almost simple in the sense that $$L\trianglelefteq G\leq Aut L$$ and L non- abelian simple and G acts simply primitive on a finite set of size n. It is shown that in general $$G^{(2)}\leq N_{Sym(\Omega)}(L)$$ with the following exceptions: Two infinite series $$(L=G_ 2(q)$$, $$q\geq 3$$, $$n=q$$ 3(q 3-1)/2 and $$L=\Omega_ 7(q)$$, $$n=q$$ 3(q 4-1)/(2,q-1)) and six exceptional examples of rank 3 and $$n\leq 276.$$
From this main result completely satisfactory results for $$k>2$$ are derived. In particular, if $$k>3$$ and $$n>24$$ and G is almost simple with socle L, then $$G^{(k)}\leq N_{Sym(n)}(L)$$. The proofs use (indirectly) the classification of finite simple groups.
Reviewer: W.Knapp

##### MSC:
 20B15 Primitive groups 20B10 Characterization theorems for permutation groups 20D05 Finite simple groups and their classification
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