I am a math hobbyist studying Joseph Gallian's "Contemorary Abstract Algebra" (Fifth Edition).

Part of Theorem 6.2 on page 124 states that if is an isomorphism then the order of an element a is equal to the order of ; that is (that is isomorphisms preserve orders).

Gallian's proof is essentially as follows:

a has order n

Gallian seems to conclude the proof at this point, but to me he has only shown (using his Th 4.1 Corollary 2) that the order ofdividesn

How do you show that the order of actually equals n?