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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Proof using derived series
We can show that the derived series of descends at least as fast as the series obtained by intersecting with the members of the derived series of .
Proof using arbitrary normal series
We can show that intersecting with any normal series for with abelian quotients gives a normal series for with abelian quotients. (We could also use a subnormal series instead of a normal series).