# N-abelian implies n(n-1)-central

From Groupprops

## Contents

## Statement

Suppose is a group and is an integer other than 0 or 1. Suppose that is a n-abelian group, i.e., the map is an endomorphism of (hence, it is a universal power endomorphism). Then, is n(n-1)-central: the inner automorphism group of has exponent dividing .

## Related facts

### Converse

- 2-central implies 4-abelian
- 3-central implies 9-abelian
- 4-central implies 16-abelian
- 6-central implies 36-abelian

### Facts about n-abelian groups

We say that a group is a n-abelian group if the power map is an endomorphism. Here are some related facts about -abelian groups.

- n-abelian iff (1-n)-abelian
- The set of for which is -abelian is termed the exponent semigroup of . It is a submonoid of the multiplicative monoid of integers.
- abelian implies n-abelian for all n
- n-abelian implies every nth power and (n-1)th power commute
- n-abelian implies n(n-1)-central
- n-abelian iff abelian (if order is relatively prime to n(n-1))
- nth power map is surjective endomorphism implies (n-1)th power map is endomorphism taking values in the center
- (n-1)th power map is endomorphism taking values in the center implies nth power map is endomorphism
- Frattini-in-center odd-order p-group implies p-power map is endomorphism
- Frattini-in-center odd-order p-group implies (mp plus 1)-power map is automorphism
- Characterization of exponent semigroup of a finite p-group
- Alperin's structure theorem for n-abelian groups

Value of (note that the condition for is the same as the condition for ) | Characterization of -abelian groups | Proof | Other related facts |
---|---|---|---|

0 | all groups | obvious | |

1 | all groups | obvious | |

2 | abelian groups only | 2-abelian iff abelian | endomorphism sends more than three-fourths of elements to squares implies abelian |

-1 | abelian groups only | -1-abelian iff abelian | |

3 | 3-abelian group means: 2-Engel group and derived subgroup has exponent dividing three | Levi's characterization of 3-abelian groups | cube map is surjective endomorphism implies abelian, cube map is endomorphism iff abelian (if order is not a multiple of 3), cube map is endomorphism implies class three |

-2 | same as for 3-abelian | (based on n-abelian iff (1-n)-abelian) |

## Facts used

## Proof

**Given**: A group such that the map is an endomorphism of . Elements (possibly equal).

**To prove**: commutes with . This is sufficient because being arbitrary, it shows that is central, and being arbitrary, it shows that has exponent dividing .

**Proof**:

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | commutes with . | Fact (1) | The map is an endomorphism of . | By the given data and Fact (1), every power commutes with every power. Treating as a power and as a power, we obtain that commutes with . | |

2 | commutes with . | Fact (1) | The map is an endomorphism of . | By the given data and Fact (1), every power commutes with every power. Treating as a power and as a power, we obtain that commutes with . | |

3 | The centralizer in of contains the elements and . Since the centralizer of any element is a subgroup, it also contains the quotient , and hence commutes with . | Steps (1), (2) | The first part follows from Steps (1) and (2). The rest is self-explanatory. |