# Difference between revisions of "(n-1)th power map is endomorphism taking values in the center implies nth power map is endomorphism"

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==Statement== | ==Statement== | ||

− | Suppose <math>G</math> is a [[group]] and <math>n</math> is an integer such that the map <math>x \mapsto x^{n-1}</math> is an [[endomorphism]] of <math>G</math> and <math>x^{n-1}</math> is in the [[center]] of <math>G</math> for all <math>x</math> in <math>G</math>. Then, the map <math>x \mapsto x^n</math> is also an [[endomorphism]] of <math>G</math>. | + | Suppose <math>G</math> is a [[group]] and <math>n</math> is an integer such that the [[fact about::universal power map|power map]] <math>x \mapsto x^{n-1}</math> is an [[endomorphism]] of <math>G</math> and <math>x^{n-1}</math> is in the [[center]] of <math>G</math> for all <math>x</math> in <math>G</math>. Then, the map <math>x \mapsto x^n</math> is also an [[endomorphism]] of <math>G</math>. |

==Related facts== | ==Related facts== | ||

+ | ===Applications=== | ||

+ | |||

+ | * [[Frattini-in-center odd-order p-group implies (mp plus 1)-power map is automorphism]] | ||

===Converse=== | ===Converse=== | ||

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'''To prove''': The map <math>x \mapsto x^n</math> is an endomorphism of <math>G</math>, i.e., <math>(gh)^n = g^nh^n</math> for all <math>g,h \in G</math>. | '''To prove''': The map <math>x \mapsto x^n</math> is an endomorphism of <math>G</math>, i.e., <math>(gh)^n = g^nh^n</math> for all <math>g,h \in G</math>. | ||

− | '''Proof''': We have the following for all <math>g,h | + | '''Proof''': We have the following for all <math>g,h \in G</math>. |

{| class="sortable" border="1" | {| class="sortable" border="1" |

## Latest revision as of 20:09, 25 February 2011

## Statement

Suppose is a group and is an integer such that the power map is an endomorphism of and is in the center of for all in . Then, the map is also an endomorphism of .

## Related facts

### Applications

### Converse

The precise converse is not true, but a partial converse is: nth power map is surjective endomorphism implies (n-1)th power map is endomorphism taking values in the center

## Proof

**Given**: A group , an integer such that the map is an endomorphism taking values in the center.

**To prove**: The map is an endomorphism of , i.e., for all .

**Proof**: We have the following for all .

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

1 | |||

2 | is an endomorphism | ||

3 | Step (2) plugged into Step (1) | ||

4 | is in the center of , so commutes with | Step (3) | |

5 | Step (4) (simplified) |