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I'd like to show the following result of group theory:

Problem: Let $G$ be a solvable group and $H\trianglelefteq G$. Then $G/H$ is solvable.

Definition: A group $G$ is said to be solvable if there is a normal series of $G$ such that the factors are abelian, that is, there exists a sequence of normal subgroups:$$G=G_0\trianglerighteq G_1\trianglerighteq \ldots \trianglerighteq G_n=\{1\},$$ such that the factors $G_{i}/G_{i+1}$ are abelians.

Sketch: Let $$G=G_0\trianglerighteq G_1\trianglerighteq \ldots \trianglerighteq G_n=\{1\},$$

a normal series of $G$ such that $G_i/G_{i+1}$ are abelians. The first try would be to consider

$$\frac{G}{H}=\frac{G_0}{H}\trianglerighteq \frac{G_1}{H}\trianglerighteq \ldots \trianglerighteq \frac{G_n}{H}.$$

Of course this makes no sense for we don't even have $H\subset G_i$ for some $i$. But we can get subgroups of $G$ containing $H$ considering $HG_i$. Since $H\trianglelefteq G$ we have $HG_i\leq G$ for all $i$. So it is natural to consider the sequence: $$G=HG_0\geq HG_1\geq \ldots \geq HG_n=H\geq \{1\}.$$

It is not hard to verify this is a normal series of $G$. I conjecture the normal series for $G/H$ will arise from this but I still can't see how.

Can anyone help me finishing this proof?

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    $\begingroup$ Your almost there. Use $HG_i/H$. What can you say about $(HG_i/H)/(HG_{i+1}/H)$? $\endgroup$ – Hagen von Eitzen Oct 10 '14 at 21:23
  • $\begingroup$ Is isomorphic to $HG_i/HG_{i+1}$.. $\endgroup$ – PtF Oct 10 '14 at 21:34
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Consider the series $$ G/H=(HG_0)/H \ge (HG_1)/H \ge \dots\ge (HG_n)/H=\{1\}. $$

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  • $\begingroup$ The problem is that the last subgroup should be identity of the group $G/H$..I had already an idea, the homomorphic image of a normal series is a normal series, so it suffices applying the natural projection on the series $G=HG_0\geq HG_1\geq \ldots \geq HG_n=H\geq \{1\}$.. $\endgroup$ – PtF Oct 10 '14 at 21:29
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    $\begingroup$ The last group in the series I wrote, $(HG_n)/H$, is the identity of $G/H$. But yes, projecting the series you wrote in original post (without the extra $\{1\}$ at the end) will give the normal series you want. The only problem is proving the quotients are abelian... $\endgroup$ – Mike Earnest Oct 10 '14 at 21:38
  • $\begingroup$ Sorry I missed that.. $\endgroup$ – PtF Oct 10 '14 at 21:40
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Homomorphic image of a solvable group is solvable. Consider natural homomorphism from G to G/H.

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  • $\begingroup$ Presumably, the point of this problem is to show that homomorphic image of a solvable group is solvable. $\endgroup$ – ronno Oct 11 '14 at 3:23

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