Not isomorphic, different order

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In summary, the conversation discusses a homework problem that involves showing that the multiplicative group Z5 is not isomorphic to the multiplicative group Z8. The solution involves finding the units of both groups and showing that they have different elements of order. The conversation also discusses the challenges the individual is facing in understanding the material and the advice given to review the basics before attempting the homework problem.
  • #1
POtment
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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.

The Attempt at a Solution


Once again, I'm not even sure how to begin this one.
 
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  • #2
The "multiplicative group [itex]\mathbb{Z}_n[/itex]" has as elements those elements of [itex]\mathbb{Z}_n[/itex] that are units (that is, those elements that have multiplicative inverses). It should be easy to find the units of [itex]\mathbb{Z}_5[/itex] and [itex]\mathbb{Z}_8[/itex]. Why not try to start with that?
 
  • #3
I'm really lost in this class. I'm not even sure how to do that...
 
  • #4
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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  • #5
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.
 
  • #6
POtment said:
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 [itex]\mathbb{Z}_n[/itex], 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.
 
  • #7
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!)
 

1. What does it mean for two structures to be "not isomorphic"?

"Not isomorphic" means that two structures, such as graphs or groups, cannot be mapped onto each other in a way that preserves their structural properties. In other words, they are fundamentally different from each other and cannot be considered equivalent.

2. How can we determine if two structures are "not isomorphic"?

To determine if two structures are "not isomorphic", we can use various techniques such as graph isomorphism algorithms or group isomorphism tests. These methods involve comparing the structural properties of the two structures and looking for differences that cannot be reconciled.

3. What is the significance of two structures being "not isomorphic"?

The significance of two structures being "not isomorphic" is that they represent distinct mathematical objects with unique properties and relationships. This allows us to study and understand different aspects of mathematics and the world around us through these diverse structures.

4. Can two structures that are "not isomorphic" have similarities?

Yes, two structures that are "not isomorphic" can still have similarities. This could be in terms of their individual elements or certain structural patterns. However, these similarities are not enough to consider the structures equivalent or isomorphic.

5. How does the concept of "not isomorphic" apply to other fields of science?

The concept of "not isomorphic" applies to various fields of science such as biology, chemistry, and physics. In biology, for example, two species may have similar traits or characteristics, but they are not isomorphic as they have different genetic makeups. Similarly, in chemistry, two molecules may have similar properties, but they are not isomorphic as their atomic arrangements are different.

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