# Find generators of a group in GAP

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.

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May I ask what is your group? Thanks – Babak S. 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? – Babak S. 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. – Babak S. Apr 20 '13 at 12:31

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]

[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.

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Tried, but failed,:( – Easy Apr 20 '13 at 14:33
@Easy, what does fail in your case - getting generators or viewing the documentation? – Alexander Konovalov Apr 20 '13 at 14:59
I've now added some hints how one could try to find answers to similar questions using the GAP help system - hope this will be useful. – Alexander Konovalov Jan 31 '15 at 21:32

I have found the command:

GeneratorsOfGroup( G );

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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"

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Haha, that's exactly what I did! – Easy Apr 20 '13 at 16:01
Really! Nice to hear that, Easy. ;-) – Babak S. Apr 20 '13 at 16:02
Nice display! I need to start using GAP again. After all, I have a dear friend to turn to if I need to learn anything! – amWhy Apr 21 '13 at 1:11
@amWhy: Easy made me to search and to learn the final line, Amy. Thanks Easy for this question. It deserves more than +1 for me. – Babak S. Apr 21 '13 at 7:53