(n-1)th power map is endomorphism taking values in the center implies nth power map is endomorphism
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
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|
|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)|