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I'm trying to prove that if $H$ is a normal subgroup of a group $G$ such that $H$ and $G/H$ are finitely generated, then G is finitely generated also. I'm trying to find a finite set $X$ such that $G$ is generated by $X$, but I have no ideal how to find this set using the finite generator sets of $H$ and $G/H$.

I need help

Thanks in advance

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Can you do this if instead of groups they are vector spaces? –  Mariano Suárez-Alvarez Apr 28 '13 at 20:43
    
@MarianoSuárez-Alvarez I think yes –  user42912 Apr 28 '13 at 20:58
    
Well: exactly the same argument works for groups. –  Mariano Suárez-Alvarez Apr 28 '13 at 20:59
    
But what is $G/H$ in vector spaces algebra? –  user42912 Apr 28 '13 at 21:01
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If you don't now what that quotient is for vector spaces, how can you do this for vector spaces? –  Mariano Suárez-Alvarez Apr 28 '13 at 21:05
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1 Answer 1

up vote 2 down vote accepted

Hints: we're given

$$H=\langle\,h_1,\ldots,h_k\,\rangle\;,\;\;G/H:=\langle\,g_1H,\ldots,g_nH\,\rangle$$

Remember now that for all $\,x\in G\,$ there exist unique $\,1\le i_x\le n\,$ and unique $\,h_x\in H\,$ s.t. $\,x=g_{i_x}h_x\,$ and, of course, then $\,x\in g_{i_x}H\,$ , so...

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How does the normality of H come into play? –  user41442 Apr 29 '13 at 0:10
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Otherwise you have no quotient group at all,@user41442... –  DonAntonio Apr 29 '13 at 1:44
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