Abstract
This paper is a contribution to the study of the automorphism groups of - designs. Let be - design and Aut a block transitive and a point primitive. If is unsolvable, then Soc, the socle of , is not .
1. Introduction
A design is a pair consisting of a finite set of points and a collection of of , called blocks, such that any 2-subsets of are contained in exactly one block. We will always assume that .
Let be a group of automorphisms of a design . Then is said to be block transitive on if is transitive on and is said to be point transitive (point primitive) on if is transitive (primitive) on . A flag of is a pair consisting of a point and a block through that point. Then is flag transitive on if is transitive on the set of flags.
The classification of block transitive designs was completed about thirty years ago (see [1]). In [2], Camina and Siemons classified designs with a block transitive, solvable group of automorphisms. Li classified designs admitting a block transitive, unsolvable group of automorphisms (see [3]). Tong and Li [4] classified designs with a block transitive, solvable group of automorphisms. Han and Li [5] classified designs with a block transitive, unsolvable group of automorphisms. Liu [6] classified (where ) designs with a block transitive, solvable group of automorphisms. In [7], Han and Ma classified designs with a block transitive classical simple group of automorphisms.
This paper is a contribution to the study of the automorphism groups of designs. Let be design and a block transitive and a point primitive. We prove the following theorem.
Main Theorem. Let be design and a block transitive and a point primitive. If is unsolvable, then .
2. Preliminary Results
Let be a design defined on the point set and suppose that is an automorphism group of that acts transitively on blocks. For a design, as usual, denotes the number of blocks and denotes the number of blocks through a given point. If is a block, denotes the setwise stabilizer of in and is the pointwise stabilizer of in . Also, denotes the permutation group induced by the action of on the points of , and so .
Lemma 1 (see [8]). Let , , , then every maximal subgroup of is conjugate to one of the following: (1); (2); (3); (4); (5), if ; (6); (7); (8); (9); (10); (11); (12), where and be prime.
Lemma 2 (see [9]). Let be an exceptional simple group of Lie type over , and let be a group with . Suppose that is a maximal subgroup of not containing , then one of the following holds: (1); (2), or , if ; (3) is a parabolic subgroup of .
Lemma 3 (see [7]). Let and be a group and a design, and be block transitive and point primitive but not flag transitive. Let . Then where and is the length of the longest suborbit of on .
3. Proof of the Main Theorem
Proposition 4. Let be design and let be block transitive and point primitive but not flag transitive, then .
Proof. Let . Obviously, Since , we get . Otherwise, , by [9], is flag transitive, a contradiction. Thus, .
Proposition 5. Let be design and let be block transitive and point primitive but not flag transitive and be even. If be unsolvable, then .
Proof. Let . Since is unsolvable, then the structure of and the rank and subdegree of do not occur: Otherwise, is odd and . We have and are odd. Since and , then is odd and is also odd, a contradiction with be even. Thus, . By Lemma 3, By Proposition 4,
Now we may prove our main theorem.
Suppose that , then . We have , where , the outer automorphisms group of which may be generated by an automorphism of field. By [8], we may assume that is an automorphism of field. Set , then . Obviously, . By [10] and , is not flag transitive. Since is point primitive, () is the maximal subgroup of and is block transitive in . Hence, satisfies one of the three cases in Lemma 2. We will rule out these cases one by one.
Case 1 (). By Proposition 5, we have an upper bound of , We get that is, Thus, or and or . Since is odd, then contains a Sylow 2-subgroup of . Clearly is contained in some maximal subgroups of . By Lemma 1, , where (). Then or . By Proposition 4 and rather long and repetitive numerical calculations, we get a contradiction.
Case 2 (, or , if ). Obviously, and or . By Proposition 4, we have or , a contradiction.
Case 3 ( is a parabolic subgroup of ). By Lemma 1, the parabolic subgroup of is conjugate to or . Then the order of parabolic subgroup is or . We get or . By Proposition 4, we have or . But Obviously, or , a contradiction.
This completes the proof the Main Theorem.
Acknowledgments
This work is supported by the NNSFC (Grant no. 11271208) and the Scientific Reaserch Fund of Heilonjiang Provincial Education Departement (Grant no. 11553116).