blocks of permutation groups

Throughout this article, A is a set and G is a permutation groupMathworldPlanetmath on A.

A block is a subset B of A such that for each σG, either σB=B or (σB)B=, where σB={σ(b)bB}. In other words, if σB intersects B, then σB=B.

Note that for any such permutation group, each of , A, and every element of A is a block. These are called trivial blocks.

It is obvious that if HG are permutation groups on A, then any block of G is also a block of H.

Blocks are closed under finite intersectionDlmfMathworld:


If B1,B2A are blocks of G, then B=B1B2 is a block of G.


Choose σG. Note that σ(B1B2)=(σB1)(σB2). Thus if (σB)B, then


is nonempty, and thus σBiBi for i=1,2. But B1 and B2 are blocks, so that σBi=Bi for i=1,2. Thus


and B is a block. ∎

We show, as a corollary to the following theorem, that blocks themselves are permuted by the action of the group.


If HG are permutation groups on A, BA is a block of H, and σG, then σB is a block of σHσ-1.


Choose τH and assume that


Then, applying σ-1 to this equation, we see that


But B is a block of H, so τB=B. Multiplying by σ, we see that


and thus


and the result follows. ∎


If B is a block of G, σG, then σB is also a block of G.


Set G=H in the above theorem. ∎


If B is a block of G, σG, then B and σB are conjugate blocks. The set of all blocks conjugate to a given block is a block system.

It is clear from the fact that B is a block that conjugate blocks are either equal or disjoint, so the action of G permutes the blocks of G. Then if G acts transitively on A, the union of any nontrivial block and its conjugates is A.


If G is finite and G acts transitively on A, then the size of a nonempty block divides the order of G.


Since G acts transitively, A is finite as well. All conjugates of the block have the same size; since the action is transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, the union of the block and all its conjugates is A. Thus the size of the block divides the size of A. Finally, by the orbit-stabilizer theorem, the order of G is divisible by the size of A. ∎

Title blocks of permutation groups
Canonical name BlocksOfPermutationGroups
Date of creation 2013-03-22 17:19:05
Last modified on 2013-03-22 17:19:05
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 15
Author rm50 (10146)
Entry type Topic
Classification msc 20B05
Defines trivial block
Defines block
Defines block system
Defines conjugate block