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

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Statement

Suppose G is a group and n is an integer such that the power map x \mapsto x^{n-1} is an endomorphism of G and x^{n-1} is in the center of G for all x in G. Then, the map x \mapsto x^n is also an endomorphism of G.

Related facts

Proof

Given: A group G, an integer n such that the map x \mapsto x^{n-1} is an endomorphism taking values in the center.

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

Proof: We have the following for all g,h \in G.

Step no. Assertion/construction Given data used Previous steps used
1 (gh)^n = (gh)(gh)^{n-1}
2 (gh)^{n-1} = g^{n-1}h^{n-1} x \mapsto x^{n-1} is an endomorphism
3 (gh)^n = ghg^{n-1}h^{n-1} Step (2) plugged into Step (1)
4 (gh)^n = gg^{n-1}hh^{n-1} g^{n-1} is in the center of G, so commutes with h Step (3)
5 (gh)^n = g^nh^n Step (4) (simplified)