Isoclinic groups have same nilpotency class

From Groupprops
Jump to: navigation, search

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

Similar facts about isoclinic groups

Proof

This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format

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.