transitive actions are primitive if and only if stabilizers are maximal subgroups
First claim that if is transitive on and is a block (http://planetmath.org/BlockSystem) with , then is a subgroup of containing . It is obvious that is a subgroup, since
But also, if for , then , so and thus since is a block system and thus . This proves the claim.
To prove the theorem, note that for each , there is by the claim a correspondence between containing and subgroups of containing . Thus, is primitive on if and only if all blocks are either of size or equal to , if and only if any group containing is either itself or , if and only if for all , is maximal in . ∎
|Title||transitive actions are primitive if and only if stabilizers are maximal subgroups|
|Date of creation||2013-03-22 17:19:07|
|Last modified on||2013-03-22 17:19:07|
|Last modified by||rm50 (10146)|