Isoclinic groups have same nilpotency class

Statement

Suppose $G_{1}$ and $G_{2}$ are 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.

Related facts

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

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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.

Step no.	Assertion/construction	Facts used	Previous steps used
1	$G_{1}$ is nilpotent if and only if its inner automorphism group is nilpotent, and if so, the nilpotency class of $G_{1}$ is one more than the nilpotency class of its inner automorphism group (unless $G_{1}$ is trivial).	definition of nilpotent group, via the upper central series.	--
2	$G_{2}$ is nilpotent if and only if its inner automorphism group is nilpotent, and if so, the nilpotency class of $G_{2}$ is one more than the nilpotency class of its inner automorphism group (unless $G_{2}$ is trivial).	definition of nilpotent group, via the upper central series.	--
3	The inner automorphism group of $G_{1}$ is isomorphic to the inner automorphism group of $G_{2}$.	definition of isoclinism	$G_{1}$ is isoclinic to $G_{2}$.
4	$G_{1}$ is nilpotent if and only if $G_{2}$ is nilpotent, and they have the same nilpotency class unless one of them is trivial.		Steps (1)-(3)
5	If either group is trivial, the inner automorphism group of both must be trivial, so both must be abelian.		Step (3).

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