{"action":"created","issue":{"url":"https://api.github.com/repos/gap-system/gap/issues/883","repository_url":"https://api.github.com/repos/gap-system/gap","labels_url":"https://api.github.com/repos/gap-system/gap/issues/883/labels{/name}","comments_url":"https://api.github.com/repos/gap-system/gap/issues/883/comments","events_url":"https://api.github.com/repos/gap-system/gap/issues/883/events","html_url":"https://github.com/gap-system/gap/issues/883","id":171061562,"number":883,"title":"There should be an operation to find subgroup fixed by an automorphism","user":{"login":"stevelinton","id":6358086,"avatar_url":"https://avatars.githubusercontent.com/u/6358086?v=3","gravatar_id":"","url":"https://api.github.com/users/stevelinton","html_url":"https://github.com/stevelinton","followers_url":"https://api.github.com/users/stevelinton/followers","following_url":"https://api.github.com/users/stevelinton/following{/other_user}","gists_url":"https://api.github.com/users/stevelinton/gists{/gist_id}","starred_url":"https://api.github.com/users/stevelinton/starred{/owner}{/repo}","subscriptions_url":"https://api.github.com/users/stevelinton/subscriptions","organizations_url":"https://api.github.com/users/stevelinton/orgs","repos_url":"https://api.github.com/users/stevelinton/repos","events_url":"https://api.github.com/users/stevelinton/events{/privacy}","received_events_url":"https://api.github.com/users/stevelinton/received_events","type":"User","site_admin":false},"labels":[],"state":"open","locked":false,"assignee":null,"assignees":[],"milestone":null,"comments":1,"created_at":"2016-08-14T14:30:56Z","updated_at":"2016-08-15T18:16:51Z","closed_at":null,"body":"If there is a natural choice of operation for this I can't find it.\r\n\r\nSomething like FixedSubgroup(G, alpha) should exist and at least give \"No Method Found\". \r\n\r\nA subsidiary issue is what would be a sensible set of basic methods.\r\n\r\nNot just for groups, actually -- makes sense for arbitrary magmas, I think. "},"comment":{"url":"https://api.github.com/repos/gap-system/gap/issues/comments/239882219","html_url":"https://github.com/gap-system/gap/issues/883#issuecomment-239882219","issue_url":"https://api.github.com/repos/gap-system/gap/issues/883","id":239882219,"user":{"login":"fingolfin","id":241512,"avatar_url":"https://avatars.githubusercontent.com/u/241512?v=3","gravatar_id":"","url":"https://api.github.com/users/fingolfin","html_url":"https://github.com/fingolfin","followers_url":"https://api.github.com/users/fingolfin/followers","following_url":"https://api.github.com/users/fingolfin/following{/other_user}","gists_url":"https://api.github.com/users/fingolfin/gists{/gist_id}","starred_url":"https://api.github.com/users/fingolfin/starred{/owner}{/repo}","subscriptions_url":"https://api.github.com/users/fingolfin/subscriptions","organizations_url":"https://api.github.com/users/fingolfin/orgs","repos_url":"https://api.github.com/users/fingolfin/repos","events_url":"https://api.github.com/users/fingolfin/events{/privacy}","received_events_url":"https://api.github.com/users/fingolfin/received_events","type":"User","site_admin":false},"created_at":"2016-08-15T18:16:51Z","updated_at":"2016-08-15T18:16:51Z","body":"Some random thoughts on this (I am pretty sure @stevelinton is aware of all of that, but I thought it might be useful to record this here nevertheless):\r\n\r\n* I'd normally refer to this as a centralizer. I.e. `Centralizer(G, alpha)`. That doesn't currently work, of course, but one might consider adding such a thin. I.e. in addition or as an alternative to the notation `FixedSubgroup(G, alpha)`\r\n\r\n* Right now, the perhaps shortest (but quite inefficient) way to to compute this is `H:=Stabilizer(G,alpha,function(pt,g) return pt^InnerAutomorphism(G,g); end);`\r\n\r\n* If alpha is an inner automorphism (i.e. `IsConjugatorIsomorphism(alpha)=true`), this can be specialized to a centralizer computation. Here is a crude example:\r\n\r\n```gap\r\ngap> G:=SymmetricGroup(20); A:=AutomorphismGroup(G);\r\nSym( [ 1 .. 20 ] )\r\n<group with 3 generators>\r\ngap> alpha:=A.2;;\r\ngap> fixG:=Centralizer(G,ConjugatorOfConjugatorIsomorphism(alpha));\r\nGroup([ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20) ])\r\n```\r\n\r\n* A more general approach is to compute a centralizer inside the semdirect product of $G$ and $<\\alpha>$. For groups where such semidirect products are efficient (e.g. perm groups), this seems to work well, too:\r\n```gap\r\ngap> G:=SymmetricGroup(20); A:=AutomorphismGroup(G);\r\nSym( [ 1 .. 20 ] )\r\n<group with 3 generators>\r\ngap> alpha:=A.2;;\r\ngap> H:=SemidirectProduct(Subgroup(A,[alpha]),G);\r\n<permutation group with 4 generators>\r\ngap> fixH := Centralizer(Image(Embedding(H,2)), alpha^Embedding(H,1));\r\nGroup([ (10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29) ])\r\ngap> fixG := PreImage(Embedding(H,2), fixH);\r\nGroup([ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20) ])\r\n```"}}
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