Solvability is subgroup-closed

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

Suppose $G$ is a solvable group and $H$ is a subgroup of $G$. Then, $H$ is also a solvable group, and the derived length of $H$ is less than or equal to the derived length of $G$.

Related facts

Similar facts

• Solvability is quotient-closed
• Solvability is quasivarietal
• Solvability is extension-closed

Proof

Proof using derived series

We can show that the derived series of $H$ descends at least as fast as the series obtained by intersecting $H$ with the members of the derived series of $G$.

Proof using arbitrary normal series

We can show that intersecting $H$ with any normal series for $G$ with abelian quotients gives a normal series for $H$ with abelian quotients. (We could also use a subnormal series instead of a normal series).

Proof using the quasivarietal nature

The proof follows directly from the fact that solvability is quasivarietal, because quasivarietal implies subgroup-closed.