Isomorphic Groups and Isomorphisms in Group Theory | Abstract Algebra

Описание: We introduce isomorphic groups and isomorphisms. We'll cover the definition of isomorphic groups, the definition of isomorphism, an example of isomorphic groups with a group table, we'll prove two groups are isomorphic, discuss how to show two groups are not isomorphic, and finish with a few theorems. In short, an isomorphism f from a group G to a group H, is a bijection from G to H such that for all a, b in G, f(ab) = f(a)f(b). We say the isomorphism "preserves the group operation". If an isomorphism exists between G and H, then G and H are said to be isomorphic. #abstractalgebra #grouptheory

In our first example we see how the group Z3 is isomorphic to a multiplicative group of 3 elements. In our second example we see how the group of real numbers under addition is isomorphic to the multiplicative group of positive reals. To do this we use the bijection f(x) = e^x.

What are Bijective Functions?:    • Bijective Functions and Why They're Import...  
Basic Properties of Isomorphisms: (coming soon)
Group Isomorphism is an Equivalence Relation: (coming soon)
Proof of Cayley's Theorem: (coming soon)

0:00 - What is an Isomorphism?
1:45 - Definition of an Isomorphism and Isomorphic Groups
3:45 - Further Explanation of Preserving the Group Operation
4:26 - Isomorphisms are Renamings
5:08 - Example with Group Tables
7:19 - Proving two Groups are Isomorphic
11:20 - How to Show two Groups are NOT Isomorphic
12:29 - Some Theorems

Abstract Algebra Course:    • Abstract Algebra  
Abstract Algebra Exercises:    • Abstract Algebra Exercises  

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