Why doesn't the image of a group have the same cardinality as the group?

In summary: The first isomorphism theorem states that any two groups have an isomorphism that is a one-to-one correspondence.
  • #1
SMA_01
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I was doing one of the proofs for my abstract algebra class, and we had to prove that the cardinality of the image of G, [θ(G)] is a divisor lGl. I'm trying to intuitively understand why G and it's image don't necessarily have the same cardinality. I'm thinking it's because there isn't necessarily a one to one correspondence, if there is then they have the same cardinality, but if they don't then the cardinality of the image is less than the cardinality of G, am I thinking along the right track here? This isn't a homework problem, but I came across it in my work. Any insight is appreciated.

Thanks
 
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  • #2
The image of G in what function?
 
  • #3
@HallsofIvy: Sorry, should have been more clear. For θ:G → G' and it's a group homomorphism. I was writing the proof for it, and was wondering.
 
  • #4
SMA_01 said:
@HallsofIvy: Sorry, should have been more clear. For θ:G → G' and it's a group homomorphism. I was writing the proof for it, and was wondering.

What are you allowed to use? If you know that the image of a homomorphism is a subgroup and you know the order of a subgroup divides the order of the group, you're done. Otherwise you'll have to prove one or both of those facts from first principles.
 
  • #5
@SteveL27: The image would be a subgroup of G', right, not G? I was just wondering, why the image of G and G itself don't necessarily have the same cardinality? I guessed this was because the map from G to G' is not necessarily one-to-one and onto, but not sure if I'm correct.
 
  • #6
SMA_01 said:
@SteveL27: The image would be a subgroup of G', right, not G? I was just wondering, why the image of G and G itself don't necessarily have the same cardinality? I guessed this was because the map from G to G' is not necessarily one-to-one and onto, but not sure if I'm correct.

Yes of course. Sorry, little brain hiccup at my end. Nevermind what I wrote. You're right, if the homomorphism's not onto then the image will be a proper subset of G'.

(edit) Ok I owe you a better hint. Do you know the first isomorphism theorem?
 
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  • #7
We recently covered that actually, and considering this is a hint, I'm going to look into it and see if it clarifies things. Thanks!
 

FAQ: Why doesn't the image of a group have the same cardinality as the group?

1. Why is the cardinality of an image different from the cardinality of a group?

The cardinality of an image is different from the cardinality of a group because they are two different concepts. The cardinality of a group refers to the number of elements in a group, whereas the cardinality of an image refers to the number of distinct elements in the image set of a function.

2. Can the cardinality of an image ever be equal to the cardinality of a group?

Yes, the cardinality of an image can be equal to the cardinality of a group in certain cases. This can happen when the function is bijective, meaning that each element in the group has a unique corresponding element in the image set. In such cases, the cardinality of the group and the cardinality of the image will be the same.

3. How does the concept of surjectivity relate to the difference in cardinality between a group and its image?

Surjectivity is a property of a function where every element in the image set has a corresponding element in the domain. In other words, every element in the group has been mapped to in the image set. When a function is not surjective, the cardinality of the image will be less than the cardinality of the group because there are elements in the group that do not have a corresponding element in the image set.

4. Can the cardinality of an image be greater than the cardinality of a group?

Yes, the cardinality of an image can be greater than the cardinality of a group. This can happen when the function is not injective, meaning that there are elements in the domain that map to the same element in the image set. In such cases, the cardinality of the image will be greater than the cardinality of the group because there are fewer distinct elements in the group than in the image set.

5. How does the concept of injectivity affect the cardinality of a group and its image?

Injectivity is a property of a function where each element in the domain maps to a unique element in the image set. When a function is injective, the cardinality of the group and the cardinality of the image will be the same because there are no elements in the group that map to the same element in the image set. However, when a function is not injective, the cardinality of the image will be greater than the cardinality of the group because there are elements in the domain that map to the same element in the image set, resulting in a larger number of distinct elements in the image set.

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