Using MAGMA for Group Theory

Question

I've just started a PhD in Group Theory and need to use the computer programme MAGMA. I wonder if anyone could help me with a couple of (probably very basic things).

1. I need to produce a Hasse diagram for subgroups of a given group containing a given Sylow subgroup of the group. In MAGMA I can use the command Subgroups(G:OrderMultipleOf:=??) to obtain all subgroups of a group G which contain a Sylow subgroup, however is there a command I can use so that for a given group G and a Sylow subgroup S, I can produce all subgroups of G containing S.

2. As I'm very new to MAGMA, does anyone know of any good books, publications or websites aiding someone to use MAGMA for group theoretical purposes.

Thanks in advance for your help.

David

Answer 1

Although you say you'd prefer not to use GAP, producing a Hasse diagram is very easy in GAP, at least with the right packages.
You'll need the xgap GAP package; and either the xgap binaries, which requires an X Windows system (easiest done with Linux or a similar Unix-like system), or else Gap.app, which requires a Mac.
Once you have these installed, start xgap/Gap.app, and follow these steps:

• Type "GraphicSubgroupLattice(SymmetricGroup(4));"
  
• In the window that pops up, go to the Subgroups | All Subgroups menu.

The Hasse diagram of the subgroup lattice will appear.

It's also quite easy to show parts of the subgroup lattice -- essentially, you can take any list of subgroups and show the inclusion relations. To do this:

• Type "GraphicSubgroupLattice(G);" as before.
• Compute the list of subgroups you want to display. It should be the output of the last GAP command.
• Go to the Subgroups | Insert Vertices menu.

The Hasse diagram of the subposet consisting of subgroups from your list will appear.

There's probably a comparably easy way to show Hasse diagrams in MAGMA. (But I'm telling you what I know...)

Xgap is usually included with GAP, and is also available from:
http://www.gap-system.org/Packages/xgap.html
Gap.app is available from:
https://cocoagap.sourceforge.io/

Answer 2

&cat[[h: h in Conjugates(G,H`subgroup) | S subset h]: H in Subgroups(G)];

will create a list of all subgroups of $G$ containing a given group $S$. The main caveat here is that the function Subgroups(G) produces a list of representatives of conjugacy classes of subgroups of $G$. So after that you have to go through all conjugates of each such representative. Also, the elements of the list that Subgroups(G) returns are so-called records. To actually access the subgroup itself, you use the construction H`subgroup, where H is one such record. The command &cat concatenates a list of lists into one long list, just like the command &+list adds all the elements of the list and so on.

If you want to loop through this list, you can do something like

for h in &cat[[h: h in Conjugates(G,H`subgroup) | S subset h]: H in Subgroups(G)] do

...

end for;

or just have two nested loops, one over the elements of Subgroup(G), and one over the $G$-conjugates of each of them.

I am afraid there is no really good place to learn magma other than the online handbook and the people who already know it.

Answer 3

Rather than searching the online handbook for MAGMA it is worth downloading the handbook fro a website such as:

https://secure.msri.org/about/computing/docs/magma/