# N-abelian implies every nth power and (n-1)th power commute

From Groupprops

## Contents

## Statement

Suppose is a group and is an integer such that is n-abelian: the power map on is an endomorphism of . Then, for every we have .

## Related facts

### Applications

- nth power map is automorphism implies (n-1)th power map is endomorphism taking values in the center
- nth power map is endomorphism iff abelian (if order is relatively prime to n(n-1))
- n-abelian implies n(n-1)-central

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) |

### Analogues in other algebraic structures

- Multiplication by n map is a derivation iff derived subring has exponent dividing n
- Multiplication by n map is an endomorphism iff derived subring has exponent dividing n(n-1)

## Facts used

- Group acts as automorphisms by conjugation: For any , the map is an automorphism of .

## Proof

### From the power map being an endomorphism

**Given**: A group and a natural number such that is an endomorphism of .

**To prove**: For every , .

**Proof**: We denote by the map .

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

1 | For , we have | Fact (1) | By fact (1), is an automorphism of , so . Applying the definition of gives the result. | ||

2 | We have for all . | power map is endomorphism | Since the power map is an endomorphism, we have . Setting gives the desired result. | ||

3 | We get for all . In other words, every power commutes with every power. | Steps (1), (2) | Combining steps (1) and (2) gives that . Cancel a left-most and a right-most on both sides, and get . Multiply both sides on the right by to get the desired result.
Thus, every power commutes with every power. |