2-(υ,7,1)设计的可解区传递自同构群

Soluble Block-transitive Automorphism Groups of 2-(υ,7,l) Designs

设Ｇ是一个２－（υ，７，１）设计的可解区传递自同构群，则Ｇ是点－本原，且下列之一成立： （１）υ＝７ｎ，Ｇ是旗一传递的； （２）υ＝５６，Ｇ＝Ｚ５６：Ｈ，这里Ｈ是ＧＬ（６，５）的可解且不可约的子群； （３）υ＝ｐｎ，Ｇ≤ＡＬ（１，ｐｎ）．特别地，ｐ≠２且ｐｎ≡ｌ（ｍｏｄ ４２）．

Let G be a soluble block-transitive automorphism group of 2-(υ, 7, 1) design. Then G is point-primitive and one of the following holds: (l) υ = 7n and G is flag-transitive; (2) υ = 56 and G =Z56: H, where H is a soluble and irreducible subgroup of GL(6, 5); (3) υ= pn and G 5 A L(1,pn). In particular, p≠2 and pn ≡ 1 (mod 42).