Nth power map is surjective endomorphism implies (n-1)th power map is endomorphism taking values in the center
A precise converse does not hold, but the following does: (n-1)th power map is endomorphism taking values in the center implies nth power map is endomorphism. We cannot guarantee surjectivity in general.
- nth power map is endomorphism iff abelian (if order is relatively prime to n(n-1))
- nth power map is endomorphism implies every nth power and (n-1)th power commute
- Inverse map is automorphism iff abelian
- Square map is endomorphism iff abelian
- Cube map is endomorphism iff abelian (if order is not a multiple of 3)
- Cube map is automorphism implies abelian
- 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
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)
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Given: A group and an integer such that is a surjective endomorphism of .
To prove: is an endomorphism of and for all .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||for all .||Fact (1)||power map is endomorphism||Given+Fact direct|
|2||for all||power map is surjective||Step (1)||[SHOW MORE]|
|3||For any , , so the power map is an endomorphism||power map is endomorphism||Step (2)||[SHOW MORE]|
Steps (2) and (3) complete the proof.