Solvable Group - Properties

Properties

Solvability is closed under a number of operations.

• If G is solvable, and there is a homomorphism from G onto H, then H is solvable; equivalently (by the first isomorphism theorem), if G is solvable, and N is a normal subgroup of G, then G/N is solvable.
• The previous property can be expanded into the following property: G is solvable if and only if both N and G/N are solvable.
• If G is solvable, and H is a subgroup of G, then H is solvable.
• If G and H are solvable, the direct product G × 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 subvariety of the variety of groups, as they are closed under the taking of homomorphic images, subalgebras, and (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.