# 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...

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member 587159

## 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?

member 587159
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?

You should always check whether a map is well defined. I once wrote an entire answer to this exact same question of you. Maybe you didn't read it then. Let me know if it was useful:

https://www.physicsforums.com/threads/how-to-prove-that-a-function-is-well-defined.920532/

As a rule of thumbs, always check if a map is well-defined. Certainly when dealing with stuff that has more than 1 representation (equivalence classes, fractions, ...).