# Not isomorphic, different order

1. Homework Statement
Show that multiplicative group Z5 is not isomorphic to multiplicative group Z8 by showing that the first group has an element of order 4 but the second group does not.

3. The Attempt at a Solution
Once again, I'm not even sure how to begin this one.

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Tom Mattson
Staff Emeritus
Gold Member
The "multiplicative group $\mathbb{Z}_n$" has as elements those elements of $\mathbb{Z}_n$ that are units (that is, those elements that have multiplicative inverses). It should be easy to find the units of $\mathbb{Z}_5$ and $\mathbb{Z}_8$. Why not try to start with that?

I'm really lost in this class. I'm not even sure how to do that...

Hurkyl
Staff Emeritus
Gold Member
Then I posit, for the time being, you should forget about this homework problem, and instead spend some time reviewing what all those words and symbols mean. Don't work on the homework problem until you know what elements are in those groups, and are comfortable doing arithmetic with them.

(I'm not trying to be mean -- it's just that you've identified where your problem lies, so you should try and fix that problem directly)

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I understand that you aren't trying to be mean, and that's exactly what I am trying to do.

A worthless teacher and no book leaves me with few options - I was hoping someone here could help me understand that much and I could probably do the rest on my own.

No worries, I'll keep trying on my own.

Tom Mattson
Staff Emeritus
Gold Member
I'm really lost in this class. I'm not even sure how to do that...
Do it by brute force. The additive groups have 5 and 8 elements. Just roll up your sleeves and find the units of each group by multiplication. Remember, in the multiplicative group of units of $\mathbb{Z}_n$, inverses are unique. So once you find a multiplicative inverse of an element, you have found the only one.

Don't look for some "silver bullet" trick to solve the problem, just get in there and compute.

But yeah, if you don't know the meanings of the words and symbols I am using, then you should take Hurkyl's advice and read the book.

Thanks for responding guys.

I'm afraid I've led myself down the wrong path with this class. I gave up learning anything from the teacher and have been teaching myself mostly by looking up answers and learning backwards. I have no book to refer to, so I guess I'll just skip this one.

Thank you for your advice (and for taking the time, much appreciated!)