Isoclinic groups have same derived length

Statement
Suppose $$G_1$$ and $$G_2$$ are isoclinic groups. Then, the following are true:


 * $$G_1$$ is a solvable group if and only if $$G_2$$ is a solvable group.
 * If $$G_1$$ and $$G_2$$ are both solvable and nontrivial, then they have the same derived length. If either of them is trivial, the other may be nontrivial but must still be abelian (in which case we have derived lengths of zero and one).

Related facts

 * Isoclinic groups have same nilpotency class
 * Isoclinic groups have same non-abelian composition factors

Proof
Given: Isoclinic groups $$G_1$$ and $$G_2$$.

To prove: $$G_1$$ is solvable if and only if $$G_2$$ is, and if so, they have the same derived length if both are nontrivial. If either is trivial, the other may be nontrivial but must be abelian.