6

This is a question in the mathematical software called GAP:

What is the command for displaying all the generators of a given group?

I have been searching around but yet not found anything helpful, so I am hoping I will get a quick response here.

Olexandr Konovalov
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Easy
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  • May I ask what is your group? Thanks – Mikasa Apr 20 '13 at 12:01
  • @BabakS., I think there is no group specified in general, since I encountered this problem many times. In the current case, I construct the group $\mathbb{Z}_{25}\rtimes\mathbb{Z}_5$ by modulus of a free group. But I still need to find out the generators due to construction purpose. Of course I can still use some other clumsy ways to find them out, but just not so nice and handy. – Easy Apr 20 '13 at 12:10
  • Have you tried to find a clear presentation of above semidirect prodct? – Mikasa Apr 20 '13 at 12:22
  • @BabakS., the presentation comes from modulus of a free group, so the elements in the free group can not be identified as the elements in the quotient group. Or maybe I am not sure what you mean.. – Easy Apr 20 '13 at 12:26
  • Sorry. I mean $\mathbb Z_n⋊\mathbb Z_m=<a,b\mid a^n=b^m=1,bab^{-1}=a^l>,~~l^m\equiv 1~(mod~n),~~(l,n)=1$. Moreover there is a logical problem in the product above for $m=5,n=25$ since I cannot find that semidirectproduct by GAP. it shows an inconsistency. – Mikasa Apr 20 '13 at 12:31
  • @BabakS., I was using $\langle a,b|a^{25}=b^5=1,a^b=a^6\rangle$, which is same as what you got, but then? – Easy Apr 20 '13 at 12:33
  • @BabakS., semidirect product only works with associated action: e.g, SemidirectProduct(GL(2,3),GF(3)^2); – Easy Apr 20 '13 at 12:44
  • See this. I hope, till someone finds the proper codes, this helps you. :) http://math.stackexchange.com/q/178544/8581 – Mikasa Apr 20 '13 at 12:52
  • Are u on, Easy? – Mikasa Apr 20 '13 at 13:37
  • @BabakS., command has been found from the GAP forum. – Easy Apr 20 '13 at 14:30
  • I knew that. You can find it easily via GAP help. But I want to construct the group first. – Mikasa Apr 20 '13 at 14:31
  • @BabakS., I can construct the group. Even you can construct the group using the method you mentioned in the other question. – Easy Apr 20 '13 at 14:34

3 Answers3

7

I think you may have in mind GeneratorsOfGroup. Enter ?GeneratorsOfGroup in GAP to see the documentation.

Remark: It may be useful to extend this post by some hints on using the GAP help system from the GAP command line. A single question mark returns all manual sections (for GAP and packages) with the title starting from the given word, for example:

gap> ?generators
Help: several entries match this topic - type ?2 to get match [2]

[1] loops (not loaded): Generators
[2] Modules (not loaded): Generators
[3] Semigroups (not loaded): Generators
[4] Tutorial: GeneratorsOfSomething
...
[14] Reference: GeneratorsOfGroup

So, looking at entries 4 and/or 14 one would be able to find GeneratorsOfGroup.

Double question mark returns all manual sections whose title contains the given word, for example:

gap> ??Soluble                                                        
Help: several entries match this topic - type ?2 to get match [2]

[1] Reference: IsPSolubleCharacterTable
[2] Reference: IsPSolubleCharacterTableOp
[3] Reference: ComputedIsPSolubleCharacterTables
[4] Reference: Conjugacy Classes in Solvable Groups
[5] Reference: Irreducible Solvable Matrix Groups
[6] Reference: SupersolvableResiduum
[7] Reference: IsSolvableGroup
[8] Reference: IsSupersolvableGroup
[9] Reference: IsPSolvable

This also demonstrates another feature that recently appeared in GAP: the output below contains both versions, solvable and soluble. What happens is that the search system uses the internal variable TRANSATL (for "transatlantic" :) to search for the following pairs of spelling patterns:

gap> TRANSATL;
[ [ "atalogue", "atalog" ], [ "olour", "olor" ], [ "entre", "enter" ], 
  [ "isation", "ization" ], [ "ise", "ize" ], [ "abeling", "abelling" ], 
  [ "olvable", "oluble" ], [ "yse", "yze" ] ]

P.S. By default, the GAP help system displays the text version of the documentation. Using the SetHelpViewer command (see ?SetHelpViewer) one could set it to open its HTML-version in the browser, enjoying hyperlinks for navigation and MathJax support. Then one could add the call to SetHelpViewer to the gap.ini or gaprc file (see here) to use this setting for any new GAP session by default.

See also this answer for some more details.

Community
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Olexandr Konovalov
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4

I have found the command:

GeneratorsOfGroup( G );

Easy
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2

I did the group according to the presentation you noted above:

F:=FreeGroup("a","b");;
a:=F.1;;
b:=F.2;; 
G:=F/[a^25,b^5,b*a*b^(-1)*a^(-6)];;
Size(G);
                                      125
StructureDescription(G);
                                  "C25 : C5"
Mikasa
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