# First Isomorphism Theorem Question

• Punkyc7
In summary, the first isomorphism theorem states that there is an isomorphism of groups from Z_{8}\oplusZ_{2} onto Z_{4}\oplusZ_{4}. If the kernel of the homomorphism is nontrivial, then the homomorphism is not onto.
Punkyc7
I am having a hard time using or applying the theorem .

Anyways

Prove that there is no homomorphism from Z$_{8}$$\oplus$Z$_{2}$ onto Z$_{4}$$\oplus$Z$_{4}$

Im guessing its the First Isomorphsim Theorem because its in the chapter. But I am not sure how to use it.

Show the kernel of the homomorphism must be nontrivial. How might you do that?

just go through and try all the elements until you exhaust them all. Do I want to find the kernel of the first thing or the second thing?

Punkyc7 said:
just go through and try all the elements until you exhaust them all.

How can you do that?! You aren't given the homomorphism. Look at the orders of elements in the two groups.

Cant I just make one up?
And how do you work in the factor group?

Punkyc7 said:
Cant I just make one up?
And how do you work in the factor group?

You can make one up if you want to. But showing that isn't onto doesn't show another homomorphism might not be onto. If the kernel is nontrivial then it can't be onto. What's the reason for that?

So would the kernel be (8,0) and since (8,0)=(0,0) its not a onto homomorphism?

Punkyc7 said:
So would the kernel be (8,0) and since (8,0)=(0,0) its not a onto homomorphism?

I don't see any particular reason for why that would be the kernel.

Suppose that we call the kernel of the map K. Then (Z8+Z2)/K = Z4+Z4 is an isomorphism of groups by the first isomorphism theorem. What is the cardinality of Z8+Z2/K in terms of |K|? What is the cardinality of Z4+Z4?

Punkyc7 said:
So would the kernel be (8,0) and since (8,0)=(0,0) its not a onto homomorphism?

That's pretty poorly worded. (8,0)=(0,0) isn't a true statement. Call the homomorphism h. Then why is h((8,0))=(0,0)?

(Z8+Z2)/Z4+Z4 = K

so |K| = 16/16=1 so wouldn't the kernel be the identity?

Dick said:
That's pretty poorly worded. (8,0)=(0,0) isn't a true statement. Call the homomorphism h. Then why is h((8,0))=(0,0)?

because 8 mod 4 is zero and 0 mod 4 is zero

Punkyc7 said:
because 8 mod 4 is zero and 0 mod 4 is zero

There's a kernel of truth in there. Now if you would just state it in terms that involve group theory that might be helpful. Why are you talking about mod 4?

we are mapping over to Z4

Punkyc7 said:
we are mapping over to Z4

No you aren't! You are mapping to Z4+Z4.

yes but component wise we are mapping each thing to Z4, is that the wrong way to look at it ?

Last edited:
Hint: One of the groups has an element of order ______ but the other one ______.

aren't both groups order 16 because 8*2 =16 and 4*4=16?

Wait Z8 has an element of order 8 and Z4 has an element of at most 4. Is that right... if it is, I'm not sure if that is helpful

Punkyc7 said:
yes but component wise we are mapping each thing to Z4, is that the wrong way to look at it ?

It's the wrong way to look at it. You are throwing out numbers like you know what the homomorphism is and you are just cranking out arithmetic. You don't know what the homomorphism is. You can't do that. This is a course in group theory, not arithmetic. Group theory is an abstraction of arithmetic. Use group theory terminology to express yourself.

Punkyc7 said:
Wait Z8 has an element of order 8 and Z4 has an element of at most 4. Is that right... if it is, I'm not sure if that is helpful

That's very helpful. Now use it. It's your job to figure out how to use it.

## 1. What is the First Isomorphism Theorem?

The First Isomorphism Theorem is a fundamental theorem in abstract algebra that states that if there is a homomorphism (structure-preserving map) between two groups, then the quotient group (the group formed by taking the cosets of the kernel of the homomorphism) is isomorphic to the image of the homomorphism.

## 2. How is the First Isomorphism Theorem used in mathematics?

The First Isomorphism Theorem is used to prove the isomorphism of groups and to classify groups according to their structure. It is also used to simplify calculations and proofs in abstract algebra and other fields of mathematics.

## 3. What are the conditions for the First Isomorphism Theorem to hold?

The First Isomorphism Theorem holds when there is a homomorphism between two groups, and the kernel of the homomorphism is a normal subgroup of the domain group. Additionally, the image of the homomorphism must be a subgroup of the codomain group.

## 4. Can the First Isomorphism Theorem be applied to other mathematical structures?

Yes, the First Isomorphism Theorem can be generalized to other algebraic structures, such as rings, modules, and vector spaces. In these cases, the theorem states that if there is a homomorphism between two structures, the quotient structure is isomorphic to the image of the homomorphism.

## 5. What are the practical applications of the First Isomorphism Theorem?

The First Isomorphism Theorem has many practical applications in fields such as cryptography, coding theory, and computer science. It is also used in physics and chemistry to study symmetries and invariants of physical systems. Furthermore, the theorem has implications in other areas of mathematics, such as topology and geometry.

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