Editing Solvable group (section)

== Properties ==

Solvability is closed under a number of operations.

* If ''G'' is solvable, and ''H'' is a subgroup of ''G'', then ''H'' is solvable.<ref>Rotman (1995), {{Google books|id=lYrsiaHSHKcC|page=102|text=Every subgroup H of a solvable group G is itself solvable|title=Theorem 5.15}}</ref>
* If ''G'' is solvable, and there is a [[group homomorphism|homomorphism]] from ''G'' [[surjective|onto]] ''H'', then ''H'' is solvable; equivalently (by the [[Isomorphism theorem#First isomorphism theorem|first isomorphism theorem]]), if ''G'' is solvable, and ''N'' is a normal subgroup of ''G'', then ''G''/''N'' is solvable.<ref>Rotman (1995), {{Google books|id=lYrsiaHSHKcC|page=102|text=Every quotient of a solvable group is solvable|title=Theorem 5.16}}</ref>
* The previous properties can be expanded into the following "three for the price of two" property: ''G'' is solvable if and only if both ''N'' and ''G''/''N'' are solvable.
* In particular, if ''G'' and ''H'' are solvable, the [[direct product of groups|direct product]] ''G'' &times; ''H'' is solvable.

Solvability is closed under [[group extension]]:
* If ''H'' and ''G''/''H'' are solvable, then so is ''G''; in particular, if ''N'' and ''H'' are solvable, their [[semidirect product]] is also solvable.

It is also closed under wreath product:
* If ''G'' and ''H'' are solvable, and ''X'' is a ''G''-set, then the [[wreath product]] of ''G'' and ''H'' with respect to ''X'' is also solvable.

For any positive integer ''N'', the solvable groups of [[derived length]] at most ''N'' form a [[Algebraic variety|subvariety]] of the variety of groups, as they are closed under the taking of [[homomorphism|homomorphic]] images, [[subalgebra]]s, and [[Direct product of groups|(direct) products]].  The direct product of a sequence of solvable groups with unbounded derived length is not solvable, so the class of all solvable groups is not a variety.