Abstract Algebra 6.5: Automorphisms

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We define an automorphism of a group, look at an example, and use automorphisms to define two classes of groups.
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How to determine the function to be used, 3x mod 8?

jessatria
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please explain why since 2 is a multiple of 8 that 2x mod 8 does not form an isomorphism. what I mean here is, do we have a theorem that can help us conclude that for all x multiples of n(in this case n=8) do not give us an isomorphism

sfundomabaso