Imprimitive groups highly transitive on blocks

Abstract

We classify imprimitive groups acting highly transitively on blocks and satisfying conditions common in geometry. They can be realized as suitable subgroups of twisted wreath products.In contrast to the theory of primitive permutation groups, the literature for imprimitive groups appears needy. The best known construction principle for imprimitive groups is given by wreath products of two groups U andḠ.Suppose that U is a vector space and let V be the direct sum t i=1 U i of t copies of U . IfḠ is a subgroup of the symmetric group S t acting transitively on the set {U 1 , . . . , U t } of components of V and α :Ḡ U t → GL(U ) is a linear representation of the stabilizer inḠ of the component U t , then the twisted wreath product G = U wr αḠ may be seen as a group of affine mappings acting transitively on a suitable system of affine subspaces of V and having t blocks there. The abelian group T consisting of all translations is the corresponding inertial group, i.e. the normal subgroup of G leaving each block fixed. Restricting to aḠ-invariant subspace W of V , for which there is an integer m ≤ t such that the projection W −→ i m r=i 1 U r is an isomorphism with respect to any m-subset of {1, . . . , t}, we obtain a subgroup of G having the same system of blocks as G and inertial group given by the translations corresponding to W . In particular, the inertial group induces a regular action on each block and acts sharply transitively on m-tuples of independent points, i.e. points no two of which lying in the same block. If the representation α is transitive, then U is finite and the stabilizer of a block is 2-transitive on it. * Research supported by M.U.R.S.T..

A Class of Imprimitive Groups

“…because we may take g γ i , z of order 3 (see [1], Lemma 4.3.3). Besides, thanks to Proposition 2, we may replace z and z by z…”

“…where G m denotes the twisted wreath product U wr α A 4 , and we may regard G m as an imprimitive permutation group with the same point set and block set as G m (see [5] p. 86, [1] §2. 1 and §2.2).…”

“…. , i m } of {1, 2, 3, 4} (hence dim W = 4m) (see [1], §2.1.2 for details). Manifestly we have G 4 = G 4 , so we may assume m < 3 from now on.…”

Algebraic (2, 2)-transformation groups

“…Although classifications of imprimitive permutation groups appeared already at beginning of the last century (see [10]) and imprimitive actions play an important role in geometry, the corresponding literature is actually less well-developed than the one concerning primitive groups. For finite groups some classification has been done (see for instance [1], [5] and [11]). In [1] by using wreath products, the best-known construction principle to get imprimitive groups, a classification of finite imprimitive groups, acting highly transitively on blocks and satisfying conditions very common in geometry, is achieved.…”

“…For finite groups some classification has been done (see for instance [1], [5] and [11]). In [1] by using wreath products, the best-known construction principle to get imprimitive groups, a classification of finite imprimitive groups, acting highly transitively on blocks and satisfying conditions very common in geometry, is achieved.…”