The following is a proof of the primitivity of 2-transitive groups. We are considering the action of a group on a set. The relations refer to the set.
"the only binary relations which are preserved by a 2-transitive group are the empty relation, equality, inequality and the universal relation; only the second and fourth are equivalence relations."
I understand how the result follows, but I do not know why those four are the only binary relations preserved by a 2-transitive group - could someone please help? Is it to do with the fact that it would only preserve symmetric relations because it can swap any two points? but then surely there are more symmetric relations.
A group acts $2$-transitively on a set if it acts transitively on the distinct ordered pairs, that is, the ordered pairs $(x,y)$ with $x\ne y$. It follows that it also acts transitively on the matching pairs $(x,x)$. Thus, for a binary relation to be preserved by the action, it must have the same value on all distinct pairs and the same value on all matching pairs. That leaves only two values free to be chosen, for a total of $2\cdot2=4$ choices.