Solvability is subgroup-closed

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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

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.