Is there a formal way to show that $\mathbb{Z}_{4}$ and $\mathbb{U}_{10}$ are isomorphic? For the record,$$\mathbb{Z}_{4}=\{[0],[1],[2],[3]\},\text{ and}$$$$\mathbb{U}_{10}=\{[1],[3],[7],[9]\}.$$I can show this by means of a Cayley table or something of the sort, i.e., let$$\theta([0])=[1],$$$$\theta([1])=[3],$$$$\theta([2])=[9],\text{ and}$$$$\theta([3])=[7].$$Then $\theta:\mathbb{Z}_{4}\to\mathbb{U}_{10}$ is one-to-one and onto. Hence $\mathbb{Z}_{4}\approx\mathbb{U}_{10}$.

This process gets tedious for larger and larger sets, however.

Question

Is there a formal way to show that $\mathbb{Z}_{4}$ and $\mathbb{U}_{10}$ are isomorphic? For the record,$$\mathbb{Z}_{4}=\{[0],[1],[2],[3]\},\text{ and}$$$$\mathbb{U}_{10}=\{[1],[3],[7],[9]\}.$$I can show this by means of a Cayley table or something of the sort, i.e., let$$\theta([0])=[1],$$$$\theta([1])=[3],$$$$\theta([2])=[9],\text{ and}$$$$\theta([3])=[7].$$Then $\theta:\mathbb{Z}_{4}\to\mathbb{U}_{10}$ is one-to-one and onto. Hence $\mathbb{Z}_{4}\approx\mathbb{U}_{10}$.

This process gets tedious for larger and larger sets, however.

anonymous · Answer

I am not that in touch with my group theory, but all that I can think is drawing a Cayley table. Maybe check: http://people.reed.edu/~iswanson/abstractalgebra.pdf (Example 14.5) and http://math.berkeley.edu/~serganov/113/solsf07.pdf (2) May give you some hints.

anonymous · Answer

I will check those out. Thanks. :)