# N-abelian iff abelian (if order is relatively prime to n(n-1))

This article describes an easy-to-prove fact about basic notions in group theory, that is not very well-known or important in itself

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## Statement

Suppose is a finite group and is an integer such that the order of is relatively prime to . Then, the power map on is an endomorphism (i.e., is a n-abelian group if and only if is an abelian group.

## Related facts

### 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) |

### 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)

## Tightness

`Further information: Frattini-in-center odd-order p-group implies p-power map is endomorphism, Frattini-in-center odd-order p-group implies (p plus 1)-power map is automorphism`

It turns out that the condition of being relatively prime to is fairly tight for , i.e., we can find non-abelian groups of order *not* relatively prime to , as well as non-abelian groups of order *not* relatively prime to , where the power map is an endomorphism.

- The case: must be divisible either by 4 or by an odd prime. If 4 divides , the power map gives an automorphism for any non-abelian group of exponent 4, such as dihedral group:D8. If is an odd prime dividing , we can find a non-abelian Frattini-in-center -group, where the power map is an endomorphism taking values in the center. Hence, the power map is an automorphism.
- The case: must be divisible either by 4 or by an odd prime. If 4 divides , the power map is an endomorphism of any non-abelian group of exponent 4. If is an odd prime dividing , we can find a non-abelian Frattini-in-center -group, where the power map is an endomorphism taking values in the center. Hence, the power map is an endomorphism.

## Facts used

- Abelian implies universal power map is endomorphism
- nth power map is endomorphism implies every nth power and (n-1)th power commute
- kth power map is bijective iff k is relatively prime to the order

## Proof

### From abelianness to the power map being an endomorphism

This follows from fact (1).

### From the power map being an endomorphism to being Abelian

**Given**: A finite group and an integer such that the order of is relatively prime to . Further, is an endomorphism of .

**To prove**: is abelian.

**Proof**:

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

1 | We get for all . In other words, every power commutes with every power. | Fact (2) | power map is endomorphism | Given+Fact direct | |

2 | is abelian. | Fact (3) | Step (1) | [SHOW MORE] |