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