Determining whether homomorphism or not

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SUMMARY

The discussion centers on the concept of homomorphism, emphasizing that it preserves the structure of binary operations, specifically multiplication, within algebraic systems. It clarifies that a homomorphism is not limited to multiplication but can apply to other binary operations, depending on whether one is working with groups or rings. Key factors in determining homomorphism include the preservation of inverses, identity elements, and the values derived from the binary operation in relation to the algebraic structure being mapped.

PREREQUISITES
  • Understanding of binary operations in algebra
  • Familiarity with group theory and ring theory
  • Knowledge of algebraic structures and their properties
  • Concept of identity and inverse elements in mathematical operations
NEXT STEPS
  • Study the definitions and properties of homomorphisms in group theory
  • Explore ring theory and its distinctions from group theory
  • Learn about binary operations and their role in algebraic structures
  • Investigate examples of homomorphic mappings in various algebraic systems
USEFUL FOR

Mathematicians, students of abstract algebra, and anyone interested in the foundational concepts of homomorphism and its applications in group and ring theory.

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It seems to be that the key to answering this question is in how elements are multiplied in the domain group. In other words, the type of multiplication in the domain group. What do you think?
 
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Considering the definition of a homomorphism is that it preserves multiplicative structure, I imagine the key to figuring out whether it's a homomorphism lies in figuring out whether it preserves multiplicative structure.
 
Office_Shredder said:
Considering the definition of a homomorphism is that it preserves multiplicative structure, I imagine the key to figuring out whether it's a homomorphism lies in figuring out whether it preserves multiplicative structure.

Definition of a homomorphism from a general perspective doesn't necessarily mean that it preserves a multiplicative structure.

A homomorphism doesn't even have to be using multiplication as the binary operation.

It depends also if you're dealing with let's say, rings, or groups (ring theory vs group theory).

Basically, a binary operation is homomorphic if it preserves structure from one domain to another.

You can do this by checking if the inverses, identity, and values from the binary operation coincide with the algebraic structure being mapped to.
 
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