blocks of permutation groups
Throughout this article, is a set and is a permutation group on .
A block is a subset of such that for each , either or , where . In other words, if intersects , then .
It is obvious that if are permutation groups on , then any block of is also a block of .
If are blocks of , then is a block of .
Choose . Note that . Thus if , then
is nonempty, and thus for . But and are blocks, so that for . Thus
and is a block. ∎
We show, as a corollary to the following theorem, that blocks themselves are permuted by the action of the group.
If are permutation groups on , is a block of , and , then is a block of .
Choose and assume that
Then, applying to this equation, we see that
But is a block of , so . Multiplying by , we see that
and the result follows. ∎
If is a block of , , then is also a block of .
Set in the above theorem. ∎
If is a block of , , then and are conjugate blocks. The set of all blocks conjugate to a given block is a block system.
It is clear from the fact that is a block that conjugate blocks are either equal or disjoint, so the action of permutes the blocks of . Then if acts transitively on , the union of any nontrivial block and its conjugates is .
If is finite and acts transitively on , then the size of a nonempty block divides the order of .
Since acts transitively, is finite as well. All conjugates of the block have the same size; since the action is transitive, the union of the block and all its conjugates is . Thus the size of the block divides the size of . Finally, by the orbit-stabilizer theorem, the order of is divisible by the size of . ∎
|Title||blocks of permutation groups|
|Date of creation||2013-03-22 17:19:05|
|Last modified on||2013-03-22 17:19:05|
|Last modified by||rm50 (10146)|