Abstract Algebra - Isomorphism

In summary, the conversation discusses the presence of multiple subgroups isomorphic to S41 within S42. One of these subgroups, H, is identified and the question asks for the determination of σ1,...,σ42. The conversation also mentions the concept of cosets and hints at using Lagrange's Theorem to determine the number of cosets. Finally, one of the participants mentions studying alone and seeking help with solving the question.
  • #1
jz0101
2
0
1. Show that S42 contains multiple subgroups that are isomorphic to S41.
Choose one such subgroup H and find σ1,...,σ42 such that



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How can you solve this?? I am confused if anyone can help me to solve this!
 
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  • #2
What, in words, does S42 represent?
 
  • #3
I'll assume you already identified a subgroup ##H## that is isomorphic to ##S_{41}##. Think about the cosets of ##H##. How many are there? (If this is a hard question, try looking over a proof of Lagrange's Theorem).
 
  • #4
@kduna: I am studying by myself reading book. Now I saw this question which I wanted to know how to solve this question.
 

FAQ: Abstract Algebra - Isomorphism

1. What is an isomorphism in abstract algebra?

An isomorphism in abstract algebra is a bijective homomorphism between two algebraic structures. This means that the mapping preserves both the structure and the operations of the two structures being compared.

2. How is an isomorphism different from an automorphism?

An automorphism is a special type of isomorphism where the two algebraic structures being compared are actually the same structure. In other words, an automorphism is an isomorphism between a structure and itself.

3. Why is isomorphism important in abstract algebra?

Isomorphism is important in abstract algebra because it allows us to compare and study different algebraic structures with similar properties and operations. It also helps us to understand the underlying structure of these structures and identify patterns and relationships between them.

4. How do you prove that two algebraic structures are isomorphic?

To prove that two algebraic structures are isomorphic, you must show that there exists a bijective homomorphism between them. This can be done by demonstrating that the mapping preserves the structure and operations of both structures.

5. Can two non-isomorphic structures have the same cardinality?

Yes, it is possible for two non-isomorphic structures to have the same cardinality. This means that they have the same number of elements, but the way these elements are combined and operated on may be different, making them non-isomorphic.

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