The classification of finite linear spaces with flag-transitive automorphism groups of affine type
A classification of all finite linear spaces \(S\) with flag-transitive automorphisms group \(G\) of affine type is obtained. In this case the set of points of \(S\) can be identified with the vectors in a vector space \(V=V_d(p)\) of dimension \(d\) over \(F_p\) (\(p\) prime), in such a way that \(G\leq AGL(V)=AGL_d(p)\) and \(G\) contains the translation subgroup \(T\simeq (Z_p)^d\); moreover, if \(G_0\) denotes the stabilizer of the point \(0\in S\), then \(G=TG_0\) and \(G_0\) is an irreducible subgroup of \(GL_d(p)\). Example 1. (Desarguesian affine spaces). Here \(S\) is an affine space \(AG(n,q)\), where \(n\geq 2\) and \(q^n=p^d\); we have \(G_0\leq \Gamma L_n(q)\) and one of the following holds: (1) \(G\) is 2-transitive on \(V\); (2) \(n=2\), \(q=11\) or 23, and \(G\) is one of three soluble flag-transitive groups; (3) \(n=2\), \(q=9,11,19,29\) or 59, and \(G_0^\infty=SL_2(5)\); (4) \(n=2\), \(q=3\) and \(G_0=SL_2(5)\). Example 2. (non-Desarguesian affine spaces). The examples here are: (1) the Luneburg affine planes of order \(q^2\), where \(q=2^{2e+1}\geq 8\), and \(^2B_2(q)\triangleleft G_0\leq\) Aut\((^2B_2(q))\); (2) the nearfield plane of order 9: here there are seven possibilities for \(G\); (3) the Hering plane of order \(27\): here \(G_0=SL_2(13)\) and \(G\) is 2-transitive on points. Example 3. (Hering spaces). These are two flag-transitive linear spaces on \(3^6\) points, with lines of size \(3^2\); in both cases \(G_0=SL_2(13)\) and \(G\) is 2-transitive on points. Main theorem: Let \(S\) be a finite linear space and suppose \(S\) is a flag-transitive automorphisms group of \(S\) of affine type. Then either (i) \((S,G)\) is as in examples 1, 2, 3 above, or (ii) \(S\) has \(q=p^d\) points, and \(G\leq A\Gamma L_1(q)\).
scientific article; zbMATH DE number 1236152
scientific article