Chapter 8 - Primitivity

Question 1

For G acting transitively on Ω, define a block Γ<Ω.
Define a block system containing Γ
Show that any block system partitions Ω and each set in it is a block

Answer 1

A block Γ is a subset such that for any g in G, the action of g on Γ either fixes Γ (ie αg is in Γ whenever α is in Γ) or has Γg is totally distinct from Γ.
The block system containing Γ is the set:
Σ(Γ)={Γg:g is in G}
Clearly the sets cover Ω. Then use definitions to show that if two elements in the block system are not disjoint, they are equal.

Question 2

Define for any relation R on Ω what it means for R to be a G-congruence?
Show that if R is a G-congruence then the R equivalence classes form a block system, and conversely that any block system define a relation which is a G-congruence

Answer 2

R is a G correspondence if it respects the action of G, thus:
αRβ IFF (αg)R(βg)

This is pretty obvious. Just do the algebra.