# Frattini-in-center odd-order p-group implies (mp plus 1)-power map is automorphism

This article states and (possibly) proves a fact that is true for odd-order p-groups: groups of prime power order where the underlying prime is odd. The statement is false, in general, for groups whose order is a power of two.

View other such facts for p-groups|View other such facts for finite groups

## Statement

Suppose is an odd prime, and is a finite -group (i.e., an Odd-order p-group (?): a Group of prime power order (?) with an odd prime). Then, if is a Frattini-in-center group (?), i.e., if , and is any integer, then the map is an automorphism of .

Thus, it is a universal power automorphism. In the particular case that does *not* have exponent , this gives a non-identity universal power automorphism.

## Examples

For any odd prime , the smallest non-abelian examples are the groups of order . There are two such examples: prime-cube order group:U(3,p) (GAP ID ) and semidirect product of cyclic group of prime-square order and cyclic group of prime order (GAP ID ). The former has exponent , so the - power map is the identity automorphism. The latter has exponent , so that -power map is a non-identity universal power automorphism. In fact, this automorphism itself has order .

For the case , these groups become prime-cube order group:U(3,3) and semidirect product of Z9 and Z3 respectively, both of order 27.

## Related facts

- Frattini-in-center odd-order p-group implies p-power map is endomorphism
- Square map is endomorphism iff abelian
- Inverse map is automorphism iff abelian
- Cube map is endomorphism iff abelian (if order is not a multiple of 3)
- nth power map is endomorphism iff abelian (if order is relatively prime to n(n-1))

## Facts used

- Frattini-in-center odd-order p-group implies p-power map is endomorphism
- Abelian implies universal power map is endomorphism
- (n-1)th power map is endomorphism taking values in the center implies nth power map is endomorphism
- kth power map is bijective iff k is relatively prime to the order

## 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**: An odd prime , a finite -group that is a Frattini-in-center group, i.e., is elementary abelian, or equivalently, . is an integer.

**To prove**: The map is an automorphism of .

**Proof**:

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

1 | Define as the map . Then is an endomorphism of . | Fact (1) | is Frattini-in-center, and is odd. | -- | Fact+Given direct. |

2 | The image of is in . | is Frattini-in-center. | [SHOW MORE] | ||

3 | The map is an endomorphism of taking values in . | Fact (2) | Steps (1), (2) | [SHOW MORE] | |

4 | The map is an endomorphism of . | Fact (3) | Step (3) | Fact+Step direct, setting . | |

5 | The map is bijective from to . | Fact (4) | [SHOW MORE] | ||

6 | The map is an automorphism of . | Steps (4), (5) | Step-combination-direct. |