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– Section heading added by Undead Shambles (talk) 06:22, 3 June 2021 (UTC)[reply]
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)[reply]
Sorry - I should have signed in before using four tildes. Maproom (talk) 10:47, 5 March 2012 (UTC)[reply]
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)[reply]
Thank you. That is all clear now. Maproom (talk) 19:03, 14 March 2012 (UTC)[reply]
"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)[reply]
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)[reply]
"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)[reply]
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)[reply]
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)[reply]
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)[reply]