Talk:Rank 3 permutation group

Clarification on orbits

FYI

 – Section heading added by Undead Shambles (talk) 06:22, 3 June 2021 (UTC)

This definition makes no sense to me. According to the article on orbits, an orbit is something that a point of the permuted set has. But according to this article, the stabilizer of a point (a subgroup) can itself can have an orbit, and even three orbits. What is an orbit of a subgroup? 109.145.212.24 (talk) 23:16, 4 March 2012 (UTC)

Sorry - I should have signed in before using four tildes. Maproom (talk) 10:47, 5 March 2012 (UTC)

Orbits have two ingredients, the points being acted on and the group doing the acting. Keeping the same points, but shrinking the group to a subgroup turns the orbits into suborbits: orbits of subgroups. For instance the symmetric group Sym({1,2,3}) acts 3-transitively on {1,2,3} and so has a single orbit: {1,2,3}. A point stabilizer is Sym({1,2}) which has two orbits, {1,2} and {3}. Thus Sym({1,2,3}) has rank 2 and 2 suborbits, one of size 2 and one of size 1. A silly example of a rank 3 primitive permutation group is the alternating group on 3 points: it is primitive, but a point stabilizer is just the identity, so the suborbits are {1}, {2}, {3}. A less silly example is the dihedral group of order 10: a point stabilizer has orbits {1}, {2,5}, {3,4}. JackSchmidt (talk) 15:38, 14 March 2012 (UTC)

Thank you. That is all clear now. Maproom (talk) 19:03, 14 March 2012 (UTC)

Converting 4-transitive to rank 3

"its action on pairs of elements of S" means "its action on unordered pairs of distinct elements of S", right? Maproom (talk) 21:27, 20 June 2012 (UTC)

Yes. The article here is poorly phrased. If G is 4-transitive then it can take 1,2,3,4 to 1,2,x,y, as long as x,y are distinct from 1,2. Hence the stabilizer of {1,2} has orbits { {1,2} }, { {1,x},{2,y} : x,y ≠ 1,2 }, and { {x,y} : x,y ≠ 1,2 }, so G acts as a rank 3 group. The action of G on ordered pairs of non-distinct elements is not even transitive, and the action of the symmetric group of degree 4 on ordered pairs of distinct elements has a stabilizer with 7 orbits. JackSchmidt (talk) 23:01, 5 July 2012 (UTC)

Converting 4-transitive to rank 3, again

"If G is any 4-transitive group acting on a set S, then its action on pairs of elements of S is a rank 3 permutation group." Is this meant to be evident to the alert reader? If not, it needs a proof, or a citation of a proof. Maproom (talk) 08:02, 4 September 2013 (UTC)

Ok, I was not sufficiently alert. A cup of coffee fixed that. I have added a short proof, as a reference. It is slightly disconcerting that this is now the only reference in the article. Maproom (talk) 08:43, 4 September 2013 (UTC)

What is degree of group

... degree of the permutation group ...

What is degree? Is it order of group? If it is true, please, use more common word. If it is not true, please, refer to source with definition, where I can read about degree of group... Jumpow (talk) 21:26, 24 February 2018 (UTC)

OK, I found

When we talk of permutation group we are considering it as set of permutations of a specific set. The cardinality of that set is called the degree. So a subgroup 0f Sn as permutation of 1 to n has degree n.

I thihk, it must be inserted as note. Jumpow (talk) 21:31, 24 February 2018 (UTC)