Classification of the non-trivial 2-(k2,k,λ) designs, with λ|k, admitting a flag-transitive almost simple automorphism group
Section snippets
Introduction and main result
A 2- design is a pair with a set of v points and a set of blocks such that each block is a k-subset of and each two distinct points are contained in exactly λ blocks. We say is non-trivial if . All 2- designs in this paper are assumed to be non-trivial. An automorphism of is a permutation of the point set which preserves the block set. The set of all automorphisms of with the composition of permutations forms a group, denoted by . For a subgroup G of
Examples
This small section is devoted to the construction of the examples corresponding to cases (1)–(4) of Theorem 1.1. The tool used for the construction of the examples related to cases (1)–(2) is the geometry arising from the action of on , whereas those used for the construction of the examples related to cases (3)–(4) are some results about tactical configurations contained in [23] and [31] combined with some group-theoretical information about contained in [18].
The
Preliminary reductions
We first collect some useful results on flag-transitive designs. Lemma 3.1 Let be a 2- design and let b be the number of blocks of . Then the number of blocks containing each point of is a constant r satisfying the following: ; ; .
Proof See [23]. □
Lemma 3.2 If is a 2- design, with , admitting a flag-transitive automorphism group G, then the following hold: G acts point-primitively on . If x is any point of , then is a large subgroup of G. for any point
Reductions for X based on primitive prime divisors of its order
The first part of this section is devoted to the resolution of the case , which is achieved by combining some group-theoretical results on the structure of given in [34], together with some numerical constraints on the diophantine equation provided in [45].
The second part focuses on the case , where some tools are developed in order to control the structure of . More precisely, it is shown that is divisible by , the primitive part of (more details
is a large subgroup of X for
The aim of this section is to show that, case (2) of Theorem 4.6 implies that is a large subgroup of X. The proof strategy for doing so is as follows. Suppose that is not a large subgroup of X. Hence, a lower bound for is provided in Corollary 4.9. By using [28], [36] and [27] we determine the structure of . From this we derive an upper bound for . Then, we show that such a bound is in contrast with the lower bound for determined in Corollary 4.9, and hence no cases
is a large subgroup of X
The aim of this section is to show that implies that is a large subgroup of X. This, together with Theorem 5.8, allows us to treat all the cases of Theorem 4.6 simultaneously.
The proof strategy is as follows. Since , and since divides the length of each -orbit distinct from by Lemma 3.2(3), we derive that and, moreover, that contains a Singer cyclic subgroup of X for by using simple group-theoretical arguments. Then is a
Proof of Theorem 1.1
This final section is devoted to the completion of the proof of Theorem 1.1. The proof strategy is as follows. The group lies in a large maximal subgroup Y of X by Theorem 6.4. Moreover, Y is classified in [1]. We compare the information contained in [36] and [12] on the number of conjugacy classes of the subgroups of X isomorphic to Y and on the type 1 novelties, with those contained in Lemma 3.7 and in Proposition 4.2, and we obtain . Hence, is a known large maximal subgroup of X.
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