Solvability is extension-closed

This article gives the statement, and possibly proof, of a group property (i.e., solvable group) satisfying a group metaproperty (i.e., extension-closed group property)
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Statement

Suppose $G$ is a group with a normal subgroup $H$, so that $G/H$ is the quotient group. Suppose, further, that both $H$ and $G/H$ are solvable groups. Then, $G$ is also a solvable group.

Moreover, the derived length of $G$ is at most equal to the sum of the derived lengths of $H$ and $G/H$.

Related facts

Converse

• Solvability is subgroup-closed
• Solvability is quotient-closed
• Solvability is quasivarietal (implies both being subgroup-closed and quotient-closed).

Opposite facts

• Nilpotency is quasivarietal

Proof

Proof based on the subnormal series definition

The idea is as follows: we combine the subnormal series for $H$ with a series from $H$ to $G$ obtained by taking the inverse image of the subnormal series for $G/H$. The fourth isomorphism theorem (or more specifically, the fact that normality satisfies inverse image condition) guarantees that the inverse image of the subnormal series also satisfies the criterion that each member is adjacent in the next.

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