Isoclinic groups have same nilpotency class

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


 * $$G_1$$ is a nilpotent group if and only if $$G_2$$ is a nilpotent group.
 * If the groups are nilpotent and both nontrivial, then the nilpotency class of $$G_1$$ is the same as the nilpotency class of $$G_2$$. Note that if one of the groups is trivial, the other may be nontrivial but must still be abelian, giving a situation where one group has class zero and the other has class one.

Similar facts about same nilpotency class

 * Isologic groups with respect to fixed nilpotency class lower than theirs have equal nilpotency class

Similar facts about isoclinic groups

 * Isoclinic groups have same derived length
 * Isoclinic groups have same non-abelian composition factors
 * Isoclinic groups have same proportions of degrees of irreducible representations
 * Isoclinic groups have same proportions of conjugacy class sizes

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

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

Steps (4) and (5) together complete the proof.