# Isoclinic groups have same nilpotency class

From Groupprops

## Contents

## Statement

Suppose and are Isoclinic groups (?). Then, the following are true:

- is a nilpotent group if and only if is a nilpotent group.
- If the groups are nilpotent and both nontrivial, then the nilpotency class of is the same as the nilpotency class of . 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 and .

**To prove**: is nilpotent if and only if 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 | is nilpotent if and only if its inner automorphism group is nilpotent, and if so, the nilpotency class of is one more than the nilpotency class of its inner automorphism group (unless is trivial). | definition of nilpotent group, via the upper central series. | -- |

2 | is nilpotent if and only if its inner automorphism group is nilpotent, and if so, the nilpotency class of is one more than the nilpotency class of its inner automorphism group (unless is trivial). | definition of nilpotent group, via the upper central series. | -- |

3 | The inner automorphism group of is isomorphic to the inner automorphism group of . | definition of isoclinism | is isoclinic to . |

4 | is nilpotent if and only if 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.