(n-1)th power map is endomorphism taking values in the center implies nth power map is endomorphism
From Groupprops
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 |
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1 | ![]() |
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2 | ![]() |
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3 | ![]() |
Step (2) plugged into Step (1) | |
4 | ![]() |
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Step (3) |
5 | ![]() |
Step (4) (simplified) |