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As we know from Galois theory, an irreducible polynomial is soluble in radicals if and only if its Galois group is solvable. However, solvable groups seem to have an importance in group theory far beyond their implications for polynomial equations. For example, much effort was expended on proving the Feit–Thompson theorem, which is one of the pieces of the classification theorem, but only its corollary, that all finite simple groups of odd order are cyclic, is required for the classification, and perhaps (I do not know) this could have been proven without using the notion of solvability.

Why are solvable groups such an important subset of groups that so much research has been dedicated to their properties?

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    $\begingroup$ You have the story of the Feit-Thompson theorem backwards: what they actually proved is that all finite simple groups of odd order are cyclic, and then the solvability of all odd order groups was a corollary of that. $\endgroup$
    – Eric Wofsey
    Jul 10, 2018 at 6:51
  • $\begingroup$ @EricWofsey Thanks for that remark. I wonder then why the corollary on solvability is the form of the theorem typically stated. It seems to imply that the solvability is somehow important. $\endgroup$
    – Brian Bi
    Jul 10, 2018 at 6:57
  • $\begingroup$ Well, that's the form they chose to state it in their original paper, and that form perhaps sounds a little more impressive. But it is easy to see the two forms are equivalent. $\endgroup$
    – Eric Wofsey
    Jul 10, 2018 at 7:04
  • $\begingroup$ @BrianBi Because the solvability applies to all finite groups of odd order. Cyclicality can only be said about finite simple groups of odd order. $\endgroup$
    – Jack M
    Jul 10, 2018 at 7:29

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I trust you'll grant that abelian groups are important and deserve plenty of research.

Abelian groups have these properties (among others):

  1. Subgroups of abelian groups are abelian,

  2. Quotient groups of abelian groups are abelian.

But if $N$ is normal in $G$, and both $N$ and $G/N$ are abelian, that doesn't guarantee $G$ is abelian.

Solvable groups have these properties:

  1. Subgroups of solvable groups are solvable,

  2. Quotient groups of solvable groups are solvable,

  3. If $N$ is normal in $G$, and both $N$ and $G/N$ are solvable, then $G$ is solvable.

In a sense, solvability is inherited, both going down and going up, so it's a nicer property than commutativity.

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    $\begingroup$ In terms of making math problems easy to solve, the "nicest" property of a group is "being the trivial group", but that's not considered a very important property... $\endgroup$
    – Jack M
    Jul 10, 2018 at 7:26
  • $\begingroup$ Sure it is, @Jack, just as zero is the most important number, the one-element group is the most important group. $\endgroup$
    – Gerry Myerson
    Jul 10, 2018 at 7:29

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