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Let $G$ be a non-simple finite group with minimal number of generators $d(G)=n\geq 2$.

Does $G$ necessarily have a non-trivial quotient $H$ with $d(H)<n$?

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  • $\begingroup$ For $n=1$, how could we have $d(H)<1$? $\endgroup$ Jun 26, 2017 at 8:42
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    $\begingroup$ Even ignoring the cyclic case, one can take say $G=SL(2,5)$, which is 2-generated, but has no cyclic quotient. $\endgroup$
    – verret
    Jun 26, 2017 at 10:09
  • $\begingroup$ @DietrichBurde The trivial group $H$ has $d(H)=0$. $\endgroup$
    – Derek Holt
    Jun 26, 2017 at 10:49
  • $\begingroup$ @DerekHolt The OP wanted $H$ to be non-trivial. $\endgroup$ Jun 26, 2017 at 11:42
  • $\begingroup$ @Dietrich Burde You're right of course, I meant $n\geq 2$. I edited the question. $\endgroup$
    – ChanaG
    Jun 26, 2017 at 14:42

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