# Showing that cyclic groups of the same order are isomorphic

## Homework Statement

Prove that any two cyclic groups of the same finite order are isomorphic

## The Attempt at a Solution

So I began by looking at the map $\phi : \langle x \rangle \to \langle y \rangle$, where $\phi (x^k) = y^k$. So, I went through and showed that this is indeed an isomorphism. But when I looked at the proof in the book, is said that you first must show that this map $\phi$ is well-defined. My question here is when do I know when I should show explicitly whether a map is well-defined or not? To me it seemed relatively obvious, so I wouldn't have thought to...

Last edited:

Related Calculus and Beyond Homework Help News on Phys.org
Math_QED
Homework Helper
2019 Award

## Homework Statement

Prove that any two cyclic groups of the same finite order are isomorphic

## The Attempt at a Solution

So I began by looking at the map $\phi : \langle x \rangle \to \langle y \rangle$, where $\phi (x^k) = \phi (y^k)$. So, I went through and showed that this is indeed an isomorphism. But when I looked at the proof in the book, is said that you first must show that this map $\phi$ is well-defined. My question here is when do I know when I should show explicitly whether a map is well-defined or not? To me it seemed relatively obvious, so I wouldn't have thought to...
I think you made a mistake in the map.

Shouldn't it be:

$x^k \mapsto y^k$?

The problem here is that maybe $x^k = x^l$ for $k \neq l$. In that case, you have to check that $y^k = y^l$, or otherwise the function isn't well-defined.

This isn't as trivial as you think (well the proof is rather short but it makes use of a theorem you proved earlier).

Also, you have to check this is an isomorphism.

Is the function a homomorphism?
Is it injective?
Is it surjective?

The first two questions should be trivial, the last one follows because we have an injection from a set to another set with the same finite order.

I think you made a mistake in the map.

Shouldn't it be:

$x^k \mapsto y^k$?

The problem here is that maybe $x^k = x^l$ for $k \neq l$. In that case, you have to check that $y^k = y^l$, or otherwise the function isn't well-defined.

This isn't as trivial as you think (well the proof is rather short but it makes use of a theorem you proved earlier).

Also, you have to check this is an isomorphism.

Is the function a homomorphism?
Is it injective?
Is it surjective?

The first two questions should be trivial, the last one follows because we have an injection from a set to another set with the same finite order.
So what in general distinguishes maps that need to be checked for "well-definedness" and ones that don't? How could I tell at a glance that I should question whether one is well-defined or not? Does it have something to do with an element of the domain having more than one representation?

Math_QED