- #1

Mr Davis 97

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## Homework Statement

Show that ##\mathbb{Z}_8^*## and ##\mathbb{Z}_12^*## are isomorphic, where ##\mathbb{Z}_n^* = \{x \in \mathbb{Z} ~|~ \exists a \in \mathbb{Z}_n(ax \equiv 1~(mod~n)) \}##, and the group operation is regular multiplication.

## Homework Equations

## The Attempt at a Solution

We can see that ##\mathbb{Z}_8^* = \{1,3,5,7 \}## and ##\mathbb{Z}_8^* = \{1,5,7,11 \}##

The only possible isomorphism I can think if is a function that maps from the former to the latter such that 1 goes to 1, 3 to 5, 5 to 7, and 7 to 11. The function is obviously injective and surjective. Is the only way to show that this satisfies the homomorphism property to show that it is satisfied for each combination from the domain? This would seem to be a tedious process.