Collineation group as a subgroup of the symmetric group

Abstract

Let ψ be the projectivization (i.e., the set of one-dimensional vector subspaces) of a vector space of dimension ≥ 3 over a field. Let H be a closed (in the pointwise convergence topology) subgroup of the permutation group $\mathfrak{S}_\psi $ of the set ψ. Suppose that H contains the projective group and an arbitrary self-bijection of ψ transforming a triple of collinear points to a non-collinear triple. It is well known from [Kantor W.M., McDonough T.P., On the maximality of PSL(d+1,q), d ≥ 2, J. London Math. Soc., 1974, 8(3), 426] that if ψ is finite then H contains the alternating subgroup $\mathfrak{A}_\psi $ of $\mathfrak{S}_\psi $. We show in Theorem 3.1 that H = $\mathfrak{S}_\psi $, if ψ is infinite.

Reducts of Structures and Maximal-Closed Permutation Groups

“…If F is infinite of characteristic p with prime subfield F p , then V has as a proper non-trivial definable reduct (and group-reduct) the structure of a vector space over F p , and intermediate fields give intermediate reducts. 6. Let F = (F, +, ×) be an algebraically closed field.…”

“…In an earlier version of this paper we also asked whether the groups PGL(n, Q) (for 3 ≤ n ≤ ℵ 0 ) are maximal-closed, and more generally whether Sym(N) has any countable maximalclosed groups. However, both these questions are answered positively in [7], where it is shown that if H > PGL(n, F) is closed (F any field) and contains a permutation taking a collinear triple to a noncollinear triple, then H is the full symmetric group.…”

“…We thank two referees for careful reports which we believe have greatly benefitted the presentation of the paper. We also thank I. Kaplan for drawing our attention to [7].…”

Pairwise Nonisomorphic Maximal-Closed Subgroups of Sym(ℕ) via the Classification of the Reducts of the Henson Digraphs

“…Several related questions have recently been tackled. Independently, [3] and [9] showed that there exist nonoligomorphic maximal-closed subgroups of Sym(N), the existence of which was asked in [16]. Also, independently, [17] and [9] positively answered Macpherson's question of whether there are maximal-closed subgroups of Sym(N) of countable cardinality.…”

“…Independently, [3] and [9] showed that there exist nonoligomorphic maximal-closed subgroups of Sym(N), the existence of which was asked in [16]. Also, independently, [17] and [9] positively answered Macpherson's question of whether there are maximal-closed subgroups of Sym(N) of countable cardinality. One question that remains open is whether every proper closed subgroup of Sym(N) is contained in a maximal-closed subgroup of Sym(N), (Question 7.7 in [21] and Question 5.9 in [3]).…”