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A group is called solvable if it has a Normal Series whose Factor Groups are all Abelian . Or equivalently, if the descending normal series
:G riangleright G^{(1)} riangleright G^{(2)} riangleright \cdots,
where every subgroup is the Derived Subgroup of the previous one, ever reaches the trivial subgroup {1} of ''G''. These two definitions are equivalent, since for every group ''H'' and every normal subgroup ''N'' of ''H'', the quotient ''H''/''N'' is abelian If And Only If ''N'' includes ''H''(1).

For finite groups, an equivalent definition is that a solvable group is a group with a to Z itself, it has no composition series, but the normal series {0,Z}, with its only factor group isomorphic to Z, proves that it is in fact solvable.

In keeping with George Pólya 's dictum that "if there's a problem you can't figure out, there's a simpler problem you ''can'' figure out", solvable groups are often useful for reducing a conjecture about a complicated group, into a conjecture about a series of groups with simple structure: abelian groups (and in the finite case, cyclic groups of prime order).

All abelian groups are solvable - the Quotient ''A''/''B'' will always be abelian if ''A'' is abelian. But non-abelian groups may or may not be solvable.

A small example of a solvable, non-abelian group is the Symmetric Group ''S''3.
In fact, as the smallest simple non-abelian group is ''A''5, (the Alternating Group of degree 5) it follows that ''every'' group with order less than 60 is solvable.

The group ''S''5 is not solvable — it has a composition series {E, ''A''5, ''S''5} (and the Jordan-Hölder theorem states that every other composition series is equivalent to that one), giving factor groups isomorphic to ''A''5 and ''C''2; and ''A''5 is not abelian. Generalizing this argument, coupled with the fact that ''A''''n'' is a normal, maximal, non-abelian simple subgroup of ''S''''n'' for ''n'' > 4, we see that ''S''''n'' is not solvable for ''n'' > 4, a key step in the proof that for every ''n'' > 4 there are Polynomial s of degree ''n'' which are not solvable by radicals.

The property of solvability is in some senses inheritable, since:
  • If ''G'' is solvable, and ''H'' is a subgroup of ''G'', then ''H'' is solvable.

  • If ''G'' is solvable, and ''H'' is a normal subgroup of ''G'', then ''G''/''H'' is solvable.

  • If ''G'' is solvable, and there is a Homomorphism from ''G'' Onto ''H'', then ''H'' is solvable.

  • If ''H'' and ''G''/''H'' are solvable, then so is ''G''.

  • If ''G'' and ''H'' are solvable, the Direct Product ''G'' × ''H'' is solvable.



SUPERSOLVABLE GROUP

As a strengthening of solvability, a group ''G'' is called supersolvable (or '''supersoluble''') if it has an ''invariant'' normal series whose factors are all cyclic; in other words, if it is solvable with each ''A''''i'' also being a normal subgroup of ''G'', and each ''A''''i''+1/''A''''i'' is not just abelian, but also cyclic (possibly of infinite order). Since a normal series has finite length by definition, uncountable abelian groups are not supersolvable. In fact, all supersolvable groups are finitely generated, and an abelian group is supersolvable if and only if it is finitely generated.

If we restrict ourselves to finitely generated groups, we can consider the following arrangement of classes of groups:

: Cyclic < Abelian < Nilpotent < supersolvable < Polycyclic < '''solvable''' < Finitely Generated Group