Does the First Isomorphism Theorem apply in this case?

In summary, it is being discussed whether there exists a homomorphism from the group G, defined as integers modulo 8 direct product integers modulo 2, to the group H, defined as integers modulo 4 direct product integers modulo 4. By assuming the existence of such a homomorphism and using the first isomorphism theorem, it is being questioned what the kernel of phi would have to be. However, it is also suggested to consider them as rings instead of additive groups, where ring homomorphisms must preserve the identity.
  • #1
JasonJo
429
2
prove that there does not exist a homomorphism from G:= (integers modulo 8 direct product integers modulo 2) to H:= (intergers modulo 4 direct product integers modulo 4).

Pf:
i tried this route, assume that there is such a homomorphism. then by first isomorphism theorem, G/ker phi is isomorphic to phi(G) but what would the kernel of phi have to be in this case?
 
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  • #2
Z8 x Z2 => Z4 x Z4??
There is a homomorphism. For example phi(a, b) = (0, 2b) describes a nontrivial homomorphism, unless I've gone blind.
 
Last edited:
  • #3
Hrm. Maybe he means to consider them as rings, rather than as additive groups?

Ring homomorphisms must preserve the identity. (As well as all integer multiples of the identity...)
 

Related to Does the First Isomorphism Theorem apply in this case?

What is the First Isomorphism Theorem?

The First Isomorphism Theorem, also known as the Fundamental Homomorphism Theorem, is a fundamental result in abstract algebra that links the structure of a group or ring to its subgroups or ideals.

How does the First Isomorphism Theorem work?

The theorem states that if there is a homomorphism between two groups or rings, then the quotient group or quotient ring obtained by factoring out the kernel of the homomorphism is isomorphic to the image of the homomorphism.

What is a homomorphism?

A homomorphism is a function that preserves the algebraic structure of a group or ring. This means that the operation of the group or ring is preserved when the function is applied.

What is the kernel of a homomorphism?

The kernel of a homomorphism is the set of elements in the domain that map to the identity element in the codomain. In other words, it is the set of elements that the function "ignores" or "maps to zero".

What is the significance of the First Isomorphism Theorem?

The First Isomorphism Theorem is significant because it allows us to understand the structure of groups and rings by studying their homomorphic images. It also provides a way to classify groups and rings and identify their subgroups or ideals.

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