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A partition $P$ of a set $S$ is a set of disjoint subsets of $S$ whose union is $S$. Let $G$ be a subgroup of the symmetric group $S_n$. Define the stabilizer subgroup of $G$ for a partition $P$ of $\{1,2, \dots , n\}$ by $ G_P:= \{g \in G \ | \ gP=P \} $ where $gP:= \{ \{ga \ | \ a \in p \} \ | \ p \in P \}$.

Question: How to compute explicitly $G_P$ with GAP?

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  • $\begingroup$ Such a stabilizer should just be the direct product of smaller symmetric groups right? If we write a stabilizer element in cycle notation, each cycle should contain only elements of a set in the partition. $\endgroup$ – Rylee Lyman Aug 13 at 23:57
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    $\begingroup$ @RyleeLyman No, since a partition is defined as set of sets, the sets could be permuted, so it is (if all sets have the same size) the intersection with a wreath product. $\endgroup$ – ahulpke Aug 14 at 3:21
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Generically it would be a stabilizer

Stabilizer(G,P,OnSetsDisjointSets);

(assuming that $P$ is a set of sets in the GAP sense). IIRC this has been introduced only in recent releases, but the underlying function PartitionStabilizerPermGroup(G,P) has been around for longer. The calculation involves a backtrack search, and probably offers many possibilities for improvement.

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  • $\begingroup$ I just used your answer in the following answer: mathoverflow.net/a/338339/34538 at the end of which there is a computation my computer was not able to do. If yours can do it, I would be very interested. $\endgroup$ – Sebastien Palcoux Aug 14 at 14:51

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