# Solvability is subgroup-closed

From Groupprops

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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Get more facts about solvable group |Get facts that use property satisfaction of solvable group | Get facts that use property satisfaction of solvable group|Get more facts about subgroup-closed group property

## Contents

## Statement

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

## Related facts

### Similar facts

## Proof

### 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).

### Proof using the quasivarietal nature

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