# 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.

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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
+1 ${}{}{}{}{}{}{}$ –  Babak S. Apr 22 '13 at 14:09

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