Solve Abstract Algebra Homomorphism Problems with Step-by-Step Guidance

In summary, the conversation discusses a problem in abstract algebra involving computing quantities for a given homomorphism. The group structure under addition or multiplication must be preserved, and the ker and phi(18) values need to be found using the definition of homomorphism. The conversation also suggests using the fact that phi(1)=6 and that addition is involved.
  • #1
tinkerbell
2
0
abstract algebra ...HELP, PLZ!

THIS IS THE PROBLEM: COMPUTE THE INDICATED QUANTITIES FOR THE GIVEN HOMOMORPHISM

KER (PHI) AND PHI(18) FOR PHI: Z -> Z10 (SUBCRIPT) SUCH THAT PHI(1)=6

Can anyone please help me to solve this problem? I don't even know what it's asking for? Don't know where to start.
 
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  • #2


You might want to start out with the definition of homomorphism. In particular you should first decide whether the group structure under addition or multiplication is preserved.
 
  • #3


thanks...I've found ker=10Z, but to find fi(18), I have no clue. I don't know how to relate fi(1)=6. It must be addition
 
  • #4


Then what is [tex]\phi(1+1)[/tex]? You should be able to generate the image of all elements like this.
 

FAQ: Solve Abstract Algebra Homomorphism Problems with Step-by-Step Guidance

What is abstract algebra?

Abstract algebra is a branch of mathematics that studies algebraic structures such as groups, rings, fields, and vector spaces. It deals with the properties of these structures and the relationships between them.

Why is abstract algebra important?

Abstract algebra is important because it provides a framework for understanding and solving problems in various areas of mathematics, physics, and computer science. It also has many real-world applications, such as in coding theory, cryptography, and data analysis.

What are some key concepts in abstract algebra?

Some key concepts in abstract algebra include groups, which are sets with a binary operation that satisfies certain properties, and rings, which are sets with two binary operations that satisfy certain properties. Other important concepts include fields, vector spaces, and homomorphisms.

What are some common techniques used in abstract algebra?

Some common techniques used in abstract algebra include proving theorems using mathematical induction, finding isomorphisms between algebraic structures, and using algebraic properties to simplify equations and solve problems.

Are there any real-world applications of abstract algebra?

Yes, there are many real-world applications of abstract algebra. For example, coding theory uses abstract algebra to design error-correcting codes, cryptography uses algebraic structures to encrypt and decrypt messages, and data analysis uses algebraic techniques to analyze large datasets and identify patterns.

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