### Re: [GAP Forum] Maximal subgroups.

On Thu, Aug 31, 2017 at 09:15:51AM +, johnathon simons wrote: > According to a certain paper I've found online, it states that for > the Mathieu group M12, the triple (2A,4A,8A) is not a rigid generator > of M12: that is, G =/= where g1 is contained in 2A, > g2 is contained in 4A, g3 is contained in 8A and g1*g2*g3 = 1. In > particular, it says that (2A,4A,8A) generates a proper subgroup of M12 > as can be seen from the character tables of the maximal subgroups. Dear John Simons, dear Forum, As was explained recently on this list, you can compute the structure constant of this class triple from the character table of M12 to see that triples from these classes with product 1 exist, but that all of such triples (actually, there are several conjugacy classes of them) generate proper subgroups. > However, when I run the below algorithm (which attempts to find the > rigid generating triple) I end up "apparently" finding one. > > findNiceTriple := function(G, cls1, cls2, cls3) > local g1, g2, g3; > g1 := Representative(cls1); > for g2 in cls2 do > g3 := (g1*g2)^-1; You probably wanted to write 'G' instead of 'M12' in the following line: > if g3 in cls3 and M12 = Group(g1, g2) then > return [g1, g2, g3]; > fi; > od; > return fail; > end; (Otherwise the function looks alright.) > Then for example: > > gap> M12:=MathieuGroup(12);; > gap> cc:=ConjugacyClasses(M12);; > gap> Length(cc); > 15 > > So then to see if the triple (2A, 4A, 8A) can generate a rigid triple > of elements (g1,g1,g3) with g1*g2*g3 = 1 we have: > > gap> findNiceTriple(M12, cc[2], cc[6], cc[11]); > [ (1,8,10,12,11)(3,7,4,5,9), (2,10,7,5)(3,7,8,9), > (1,11,12,10,2,4,7,6,8,9)(3,5)] Several things look strange here: - If you had chosen the right classes, the result should be a list of three elements of order 2, 4 and 8, resp. - The second element is no permutation (where is 7 mapped to?) Use something like List(cc, c-> Order(Representative(c))); List(cc, Representative); to find the correct classes (and then you will get 'fail' from 'findNiceTriple'). Note that on different attempts the classes you want can be in different positions because GAP uses the Random function to find the conjugacy classes. > 1) If I'm not mistaken, this implies that (2A, 4A, 8A) is such a rigid > triple that generates M12? If not, could someone please clarify as to > why this is not true (is there some issue with the above algorithm If you have three conjugacy classes for which your function returns a generating triple, then you cannot conclude that there is a rigid triple (which means that the triple you found is the unique one up to conjugacy). > 2) Furthermore, if the above approach is hopeless in determining > rigid triples, could someone please inform me as why that is the As said, to prove rigidness you need to check that there is up to conjugacy only one triple with product one and that it generates the group. > case and how one can determine whether such a triple generates a > proper subgroup of M12 by simply looking at the characer tables of the > maximal subgroups? Also, as mentioned above, you do not need character tables of maximal subgroups in this case, the character table of M12 is enough. Best regards, Frank Lübeck -- /// Dr. Frank Lübeck, Lehrstuhl D für Mathematik, Pontdriesch 14/16, \\\52062 Aachen, Germany /// E-mail: frank.lueb...@math.rwth-aachen.de \\\ WWW:http://www.math.rwth-aachen.de/~Frank.Luebeck/ ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum

### Re: [GAP Forum] maximal subgroups

Please see MaximalSubgroups (I presume you should know already how to construct D8 from previous replies, so I am not giving an example). This is how to find its documentation in GAP: gap> ?MaximalSubgroups Help: several entries match this topic - type ?2 to get match [2] [1] Reference: MaximalSubgroups [2] Reference: MaximalSubgroupsLattice [3] Reference: MaximalSubgroups for groups with pcgs [4] Reference: MaximalSubgroupsTom gap> Now type ?1 etc. Also, this is useful: http://www.gap-system.org/Faq/faq.html#7.7 HTH Alexander > On 2 May 2015, at 09:37, abdulhakeem alayiwola wrote: > > how do one find the maximal subgroups of a group say D8 using gap? > ___ > Forum mailing list > Forum@mail.gap-system.org > http://mail.gap-system.org/mailman/listinfo/forum ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum

### Re: [GAP Forum] Maximal Subgroups

Hi Serkan, > I would like to know how we can compute the maximal subgroups of mathieu > group of degree 23 in gap. The maximal subgroups of M23 are stored in the AtlasRep package. The group M23 has 7 maximal subgroups. For example, the following code can be used to obtain the 4th maximal subgroup of M23 (which is isomorphic to A8, see for example the ATLAS of finite groups: http://brauer.maths.qmul.ac.uk/Atlas/v3/lookup?target=m23). gap> LoadPackage("atlasrep"); gap> N := 4; gap> gr := Group(AtlasGenerators("M23", 1, N).generators); Group([ (1,4)(5,11)(6,18)(7,10)(13,22)(14,20)(15,21)(16,19), (1,15,17,18)(2,9,22,3)(4,8,23,13)(5,7)(6,12)(11,19,14,16) ]) gap> Size(gr); 20160 See the documentation of the AtlasRep package for more information on the function AtlasGenerators. Best regards, Leandro ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum

### Re: [GAP Forum] Maximal Subgroups

Dear Serkan, On 22 January 2012 02:26, Serkan Rakin wrote: > Dear members, > > I would like to know how we can compute the maximal subgroups of mathieu > group of degree 23 in gap. please have a look at http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M23/ It contains pre-computed generators for some (if not all) maximal subgroups of M23. HTH, Dmitrii CONFIDENTIALITY: This email is intended solely for the person(s) named and may be confidential and/or privileged. If you are not the intended recipient, please delete it, notify us and do not copy, use, or disclose its content. Towards A Sustainable Earth: Print Only When Necessary. Thank you. ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum

### Re: [GAP Forum] Maximal Subgroups for O(7,3)

Thanks. I think I see what I was doing wrong. I'm still a bit perplexed as to why taking a random sample of 2- and 3-elements about 2000 times never produced a group in the other conjugacy class. Asst. Prof. Dmitrii (Dima) Pasechnik wrote: Dear Joe, I don't grok your GAP code, but the GAP generators from http://brauer.maths.qmul.ac.uk/Atlas/clas/O73/ are correct: enter them into GAP under the names given there: b11:=... #... #and then do G1:=Group(b11,b21);; G2:=Group(a11,a21);; H1:=Stabilizer(G1,1);; h:=GroupHomomorphismByImages(G1,G2,GeneratorsOfGroup(G1), GeneratorsOfGroup(G2));; gg:=List(GeneratorsOfGroup(H1),x->Image(h,x));; OrbitLengths(Group(gg),[1..1080]); [ 702, 378 ] you see that you get different classes (if they were the same, Group(gg) would fix a point) HTH, Dima if you know a representative H of G_2(3) in the original generators, a representative of the other class can be constructed by applying an outer automorphism to H. Regards, Dmitrii 2008/11/13 Joe Bohanon <[EMAIL PROTECTED]>: Sorry to those of you who get this twice. I accidentally sent it to the group pub forum first. I'm trying to get the maximal subgroups for O(7,3) and having some trouble. ATLAS 3.0 does not have them listed, but ATLAS 2.0 does have the shape and there are 7 permutation representations that can be called up by atlasrep. For each of those seven, I did the following with G set as the smallest permrep H:=Group(AtlasGenerators("O7(3)",i).generators); iso:=IsomorphismGroups(H,G); S:=Stabilizer(H,1); Then I simply ran Image(iso,S) to get the maximals corresponding to the primitive permreps. However for the two classes of G2(3), this yields conjugate maximal subgroups. In addition, I also tried to take random elements of order 2 and 3 and try to generate a G2(3), and while I was able to create many of them, none of them were out of this one conjugacy class. Am I missing something here? I don't think there is a mistake anywhere, as G2(3) is listed as having two classes in Kleidman's tables. Thanks Joe ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum

### Re: [GAP Forum] Maximal Subgroups for O(7,3)

Dear Joe, I don't grok your GAP code, but the GAP generators from http://brauer.maths.qmul.ac.uk/Atlas/clas/O73/ are correct: enter them into GAP under the names given there: b11:=... #... #and then do G1:=Group(b11,b21);; G2:=Group(a11,a21);; H1:=Stabilizer(G1,1);; h:=GroupHomomorphismByImages(G1,G2,GeneratorsOfGroup(G1), GeneratorsOfGroup(G2));; gg:=List(GeneratorsOfGroup(H1),x->Image(h,x));; OrbitLengths(Group(gg),[1..1080]); [ 702, 378 ] you see that you get different classes (if they were the same, Group(gg) would fix a point) HTH, Dima if you know a representative H of G_2(3) in the original generators, a representative of the other class can be constructed by applying an outer automorphism to H. Regards, Dmitrii 2008/11/13 Joe Bohanon <[EMAIL PROTECTED]>: > Sorry to those of you who get this twice. I accidentally sent it to the > group pub forum first. > > I'm trying to get the maximal subgroups for O(7,3) and having some trouble. > ATLAS 3.0 does not have them listed, but ATLAS 2.0 does have the shape and > there are 7 permutation representations that can be called up by atlasrep. > For each of those seven, I did the following with G set as the smallest > permrep > > H:=Group(AtlasGenerators("O7(3)",i).generators); > iso:=IsomorphismGroups(H,G); > S:=Stabilizer(H,1); > > Then I simply ran Image(iso,S) to get the maximals corresponding to the > primitive permreps. However for the two classes of G2(3), this yields > conjugate maximal subgroups. > > In addition, I also tried to take random elements of order 2 and 3 and try > to generate a G2(3), and while I was able to create many of them, none of > them were out of this one conjugacy class. > > Am I missing something here? I don't think there is a mistake anywhere, as > G2(3) is listed as having two classes in Kleidman's tables. > > Thanks > Joe ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum

### Re: [GAP Forum] Maximal Subgroups for O(7,3)

You can use: gap> g:=AtlasGroup("O7(3)",NrMovedPoints,1080);; gap> a:=AutomorphismGroup(g);; Time of last command: 16073 ms gap> h:=Stabilizer(g,1);; gap> f:=First(GeneratorsOfGroup(a),f->NrMovedPoints(Image(f,h))=1080);; gap> k:=Image(f,h);; gap> chi1:=PermutationCharacter(g,h);; Time of last command: 3636 ms gap> chi2:=PermutationCharacter(g,k);; Time of last command: 12641 ms gap> chi1=chi2; false The paper atlas mentions that these two maximals fuse in the automorphism group, so you know something like this should work. The two conjugacy classes should occur equally often if you are sampling randomly, so your method should have worked, but this is a simple direct method. On 2008-11-13, at 00:37, Joe Bohanon wrote: Sorry to those of you who get this twice. I accidentally sent it to the group pub forum first. I'm trying to get the maximal subgroups for O(7,3) and having some trouble. ATLAS 3.0 does not have them listed, but ATLAS 2.0 does have the shape and there are 7 permutation representations that can be called up by atlasrep. For each of those seven, I did the following with G set as the smallest permrep H:=Group(AtlasGenerators("O7(3)",i).generators); iso:=IsomorphismGroups(H,G); S:=Stabilizer(H,1); Then I simply ran Image(iso,S) to get the maximals corresponding to the primitive permreps. However for the two classes of G2(3), this yields conjugate maximal subgroups. In addition, I also tried to take random elements of order 2 and 3 and try to generate a G2(3), and while I was able to create many of them, none of them were out of this one conjugacy class. Am I missing something here? I don't think there is a mistake anywhere, as G2(3) is listed as having two classes in Kleidman's tables. Thanks Joe ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum

### Re: [GAP Forum] Maximal subgroups of non-representatives in lattice

Dear GAP-Forum, Vinod Valsalam asked: In the subgroup lattice produced by LatticeSubgroups(), I can find the maximal subgroup relations among conjugacy class representatives using the function MaximalSubgroupsLattice(). However, I am also interested in the maximal subgroups of all the other elements of the class; not just those of the class representatives. In GAP 3, section 7.74 of the manual describes an easy way to display this information by setting the print level to 4 or 5: This information is not any longer available via the print level, it can however be obtained easily from the maximal subgroups information by conjugating representatives. As an example I append a function that takes a subgroup lattice and writes out the lattice structure as a graph in the .dot (graphviz) format. I'd expect that this function (which will be in the next major release) is easily adapted for other purposes. Hope this helps, Alexander Hulpke -- Colorado State University, Department of Mathematics, Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA email: [EMAIL PROTECTED], Phone: ++1-970-4914288 http://www.math.colostate.edu/~hulpke # ## #F DotFileLatticeSubgroups(,) DotFileLatticeSubgroups:=function(L,file) local cls, len, sz, max, rep, z, t, i, j, k; cls:=ConjugacyClassesSubgroups(L); len:=[]; sz:=[]; for i in cls do Add(len,Size(i)); AddSet(sz,Size(Representative(i))); od; PrintTo(file,"digraph lattice {\nsize = \"6,6\";\n"); # sizes and arrangement for i in sz do AppendTo(file,"\"s",i,"\" [label=\"",i,"\", color=white];\n"); od; sz:=Reversed(sz); for i in [2..Length(sz)] do AppendTo(file,"\"s",sz[i-1],"\"->\"s",sz[i], "\" [color=white,arrowhead=none];\n"); od; # subgroup nodes, also acccording to size for i in [1..Length(cls)] do for j in [1..len[i]] do if len[i]=1 then AppendTo(file,"\"",i,"x",j,"\" [label=\"",i,"\", shape=box];\n"); else AppendTo(file,"\"",i,"x",j,"\" [label=\"",i,"-",j,"\", shape=circle]; \n"); fi; od; AppendTo(file,"{ rank=same; \"s",Size(Representative(cls[i])),"\""); for j in [1..len[i]] do AppendTo(file," \"",i,"x",j,"\""); od; AppendTo(file,";}\n"); od; max:=MaximalSubgroupsLattice(L); for i in [1..Length(cls)] do for j in max[i] do rep:=ClassElementLattice(cls[i],1); for k in [1..len[i]] do if k=1 then z:=j[2]; else t:=cls[i]!.normalizerTransversal[k]; z:=ClassElementLattice(cls[j[1]],1); # force computation of transv. z:=cls[j[1]]!.normalizerTransversal[j[2]]*t; z:=PositionCanonical(cls[j[1]]!.normalizerTransversal,z); fi; AppendTo(file,"\"",i,"x",k,"\" -> \"",j[1],"x",z, "\" [arrowhead=none];\n"); od; od; od; AppendTo(file,"}\n"); end; ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum

### Re: [GAP Forum] maximal subgroups of PSp(8,2)

Dear GAP Forum, in addition to Derek's answer, it could be mentioned that the maximal subgroups of PSp(8,2) are listed on p. 123 of the Atlas of Finite Groups. On Wed, Nov 22, 2006 at 12:22:42PM -0800, Tom Meletn wrote: > Dear GAP forum, > > Does anybody know about the structure of maximal subgroups of PSp(8,2)? > As I checked nothing is available on the ATLAS web page about it. I will be > thankful for any help. ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum

### Re: [GAP Forum] maximal subgroups of PSp(8,2)

Dear Tom, GAP Forum, There are 11 classes of these, arranged according to their Aschbacher category: Five reducibles: 2^(1+6).S(6,2) 2^(1+2+8).3.2.A6.2 2^(3+6).3.2.2^3.L(2,7) 2^(6+4).A8 3.2.S(6,2) One imprimitive: (A6 x A6).2.2.2 One semilinear: S(4,4).2 Two classical: O+(8,2).2 O-(8,2).2 Two other almost simple irreducibles: L(2,17) S10 Derek Holt. On Wed, Nov 22, 2006 at 12:22:42PM -0800, Tom Meletn wrote: > Dear GAP forum, > > Does anybody know about the structure of maximal subgroups of PSp(8,2)? > As I checked nothing is available on the ATLAS web page about it. I will be > thankful for any help. > > Tom > ___ > Forum mailing list > Forum@mail.gap-system.org > http://mail.gap-system.org/mailman/listinfo/forum ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum