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I have the following question:

Given a subgroup $ G \leq S_n $, the centralizer of $ G $ in $ S_n $ is defined as: $$ C_{S_n}(G) = \{ \sigma \in S_n : \sigma g = g \sigma \text{ for all } g \in G \}. $$

Is there a systematic method to compute $ C_{S_n}(G) $ for any given $ G \leq S_n $?

I am aware of results concerning the centralizer of a single element $ g \in S_n $, which is given by: $$ C_{S_n}(g) = \{ \sigma \in S_n : \sigma g = g \sigma \}, $$ and I have seen a solution for this case. Is there a similar approach for determining $ C_{S_n}(G) $ for any $G \leq S_n$?

N. F. Taussig
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Guy
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  • What kind of answer are you looking for; an algorithm, some clever trick/method or some theoretic result? What kind of information do you have on $G$? Maybe if you specified the solution you have seen for a single element, it would be easier to generalize. – SomeCallMeTim Sep 23 '24 at 09:49
  • This is what I have seen: https://math.stackexchange.com/questions/208790/centralizer-of-a-given-element-in-s-n I would want to extend this for the case of a group. – Guy Sep 23 '24 at 09:51
  • Great, thanks! For a first step, the centralizer of the group is just the intersection of the centralizers of each element of the basis, which you know how to compute using the link. In symbolic form, for $G=\langle g_1,g_2,\dots,g_n\rangle$, we have that $$C(G) = \bigcap_{i=1}^n C(g_i).$$ – SomeCallMeTim Sep 23 '24 at 09:58
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    @SomeCallMeTim Computation of intersections of subgroups of $S_n$ is not known to be possible in polynomial time, so that is not the best approach. – Derek Holt Sep 23 '24 at 10:21

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Yes, this can be done in polynomial time. See Section 6.1.2 of the book "Permutation Group Algorithms" by Ákos Seress.

Here is a recent paper on this topic.

Derek Holt
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  • Thanks you. Is there a way to find the group that is isomorphic to $C_{S_n}(G)$? – Guy Sep 23 '24 at 10:14
  • I don't understand what you are asking. The algorithms output generators of $C_{S_n}(G)$. The standard convention in computational group theory is that groups are defined by specifying generating sets. This allows easy (polynomial time) computation of their properties such as order and composition factors. – Derek Holt Sep 23 '24 at 10:18
  • Sorry, let me clarify. I am asking whether there is a way to know what is the group $C_{S_n}(G)$ will be as there is here: https://math.stackexchange.com/questions/208790/centralizer-of-a-given-element-in-s-n when discussing the case of a single element. – Guy Sep 23 '24 at 10:22
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    You would have to read the section in Seress' book for information on the structure of the centralizer of a subgroup. I don't think there is a straightforward description corresponding to the centralizer of an element. The case when the subgroup is transitive is easier, and I think you can say something in that case. These are all separate questions! – Derek Holt Sep 23 '24 at 10:25