Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
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
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
up vote 2 down vote accepted

Hints: we're given


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...

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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.