Dear Alejandra,
> On 09 Jan 2018, at 13:47, Alejandra Alderete <[email protected]> wrote:
>
> --
> ---------- Forwarded message ----------
> From: Alejandra Alderete <[email protected]>
> To: [email protected]
> 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
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