Subgroup of Solvable Group is Solvable/Proof 2

Theorem

Let $G$ be a solvable group.

Let $H$ be a subgroup of $G$.

Then $H$ is solvable.

Proof

Firstly we, know that a group is solvable if and only if its derived series

$\map D G = \sqbrk {G, G} \ , \ \map {D^i} G = \sqbrk {\map {D^{i - 1} } G, \map {D^{i - 1} } G}$

becomes trivial after finite iteration.

Meaning

$\map {D^j} G = \set 1$

for some finite $j$.

Now it is trivial that

$\map D H \le \map D G$

since $H$ is smaller than $G$.

Further since $\map {D^i} H$ is dominated by $\map {D^i} G$ it too has to become trivial after a finite amount of steps.

$\blacksquare$