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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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