[GAP Forum] Abelian subgroups of a Group

[Max Horn]

Dear Alejandra,

> On 09 Jan 2018, at 13:47, Alejandra Alderete <argentina.ale at gmail.com> wrote:
> 
> --
> ---------- Forwarded message ----------
> From: Alejandra Alderete <argentina.ale at gmail.com>
> To: forum at gap-web.cs.st-andrews.ac.uk
> Cc:
> Bcc:
> Date: Mon, 8 Jan 2018 09:02:34 -0300
> Subject: Abelian subgroups of a Group
> Dear Forum,
> Can someone help  me to write a program that can compute all *abelian
> subgroups* with theirs orders and theirs generators of group G generate for
> g2, g5, g9, g25, elements of S_48,  where
> 
> g2:= (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,
> 22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40
> )(41,42,43,44)(45,46,47,48),
> 
> g5:= (1,5,3,7)(2,8,4,6)(9,17,11,19)(10,20,12,18)(13,21,15,23)(14,
> 24,16,22)(25,29,27,31)(26,32,28,30)(33,37,35,39)(34,40,36,38
> )(41,45,43,47)(42,48,44,46),
> 
> g9:=(1,9,13)(2,17,22)(3,11,15)(4,19,24)(5,18,14)(6,10,21)(7,
> 20,16)(8,12,23)(25,33,41)(26,37,46)(27,35,43)(28,39,48)(29,3
> 8,42)(30,34,45)(31,40,44)(32,36,47),
> 
> g25 := (1,25)(2,31)(3,27)(4,29)(5,28)(6,32)(7,26)(8,30)(9,41)(10,47
> )(11,43)(12,45)(13,33)(14,39)(15,35)(16,37)(17,44)(18,48)(
> 19,42)(20,46)(21,36)(22,40)(23,34)(24,38),

These are not valid GAP inputs. Assuming that this is just a case of bad wrapping (note that "38" in g9 is split into "3" and "8"), and replacing the trailing commas by semicolons, we get

g2:=(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)
    (29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48);;
g5:=(1,5,3,7)(2,8,4,6)(9,17,11,19)(10,20,12,18)(13,21,15,23)(14,24,16,22)(25,29,27,31)
    (26,32,28,30)(33,37,35,39)(34,40,36,38)(41,45,43,47)(42,48,44,46);;
g9:=(1,9,13)(2,17,22)(3,11,15)(4,19,24)(5,18,14)(6,10,21)(7,20,16)(8,12,23)(25,33,41)
    (26,37,46)(27,35,43)(28,39,48)(29,38,42)(30,34,45)(31,40,44)(32,36,47);;
g25:=(1,25)(2,31)(3,27)(4,29)(5,28)(6,32)(7,26)(8,30)(9,41)(10,47)(11,43)(12,45)
    (13,33)(14,39)(15,35)(16,37)(17,44)(18,48)(19,42)(20,46)(21,36)(22,40)(23,34)(24,38);;

After entering this, we can proceed to study the group generated by these:

gap> G:=Group(g2,g5,g9,g25);
<permutation group with 4 generators>
gap> Size(G);
48
gap> StructureDescription(G);
"GL(2,3)"

So this is just GL(2,3) in disguised, and from this one could now certainly enumerate subgroups. But if you want them for this particular group, since it is very small, you can just brute force the solution using AllSubgroups():

gap> subs:=AllSubgroups(G);; Length(subs);
55
gap> abel:=Filtered(subs,IsAbelian);; Length(abel);
34

So there are 55 subgroups, of which 34 are abelian. From this you should be able to determine sizes, generators etc.

For bigger examples, more care needs to be taken. E.g. by working with conjugacy classes of subgroups:

gap> cls:=ConjugacyClassesSubgroups(G);; Length(cls);
16
gap> abelCls:=Filtered(cls, c -> IsAbelian(Representative(c)));; Length(abelCls);
8
gap> Sum(abelCls,Size);
34

So there are 16 conjugacy classes of subgroups, 8 of which are of abelian subgroups, containing a total of 34 subgroups, as expected.


Hope that helps,
Max