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Joelly
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SUMMARY
This discussion focuses on the concept of isomorphism in Abstract Algebra, specifically relating to groups defined by multiples of integers. The function f: G->H, where G represents multiples of 2 and H represents multiples of 3, is analyzed. The function is defined as f(2n) = 3n, which satisfies the criteria of being one-to-one, onto, and preserving the operation. This establishes that the two groups are isomorphic under the given function.
PREREQUISITES- Understanding of Abstract Algebra concepts
- Familiarity with group theory
- Knowledge of functions and mappings
- Basic comprehension of mathematical notation
- Study the properties of isomorphic groups in Abstract Algebra
- Explore examples of group homomorphisms
- Learn about the significance of one-to-one and onto functions
- Investigate other types of algebraic structures, such as rings and fields
Students of Abstract Algebra, mathematicians, and anyone interested in understanding group theory and isomorphism.
- #2
HOI
- 921
- 2
Are you taking an "Abstract Algebra". If so you should know what "isomorphic" means- that there is a function from one group to the other, f: G->H, that
1) it is "one-to-one"- if f(x)= f(y) the x= y.
2) it is "onto"- for every y in H there exist x in G such that f(x)= y.
3) it "preserves the operation"- f(x+ y)= f(x)+f(y).
G is defined as multiples of 2 and H is defined as multiples of 3. What about f(2n)= 3n?
1) it is "one-to-one"- if f(x)= f(y) the x= y.
2) it is "onto"- for every y in H there exist x in G such that f(x)= y.
3) it "preserves the operation"- f(x+ y)= f(x)+f(y).
G is defined as multiples of 2 and H is defined as multiples of 3. What about f(2n)= 3n?
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