Math Problem Statement
Solution
To prove Lagrange's Theorem for a finite group and its subgroup , let's proceed step by step.
Lagrange's Theorem Statement:
If is a finite group and is a subgroup of , then the order (size) of divides the order of , i.e., .
Proof Outline:
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Cosets and Partitioning of :
- Define left cosets of in as for any .
- The key idea is that the left cosets of in are disjoint and form a partition of . Each element of belongs to exactly one coset.
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Coset Properties:
- Property 1: If , then , i.e., two cosets are either identical or disjoint.
- Property 2: The size of any coset is equal to , i.e., , because multiplication by is a bijection.
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Partition Implies Divisibility:
- Since cosets partition , the total size of is the number of distinct cosets (denoted by , the index of in ) multiplied by the size of .
- Therefore, , showing that divides .
Detailed Proof:
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Establishing Coset Properties:
- If , then (this ensures cosets are well-defined).
- If , then (cosets are either disjoint or identical).
- The size of each coset is because the map for is bijective.
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Partitioning:
- The disjoint union of all cosets of gives : , where is a set of representatives for distinct cosets.
- The total number of cosets is , the index of in .
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Concluding Divisibility:
- From the partitioning, , proving that .
Final Answer for Question 4:
By the above steps, we have shown that for a finite group and a subgroup , the order of divides the order of .
Would you like a more detailed explanation of any step? Or additional examples?
Here are 5 follow-up questions you might find helpful:
- What is the significance of cosets in group theory beyond Lagrange's theorem?
- Can Lagrange's theorem apply to infinite groups?
- How do right cosets differ from left cosets, and does it affect the theorem?
- Can Lagrange's theorem provide information about possible subgroup sizes?
- What are practical examples of applying Lagrange's theorem in algebra?
Tip: Always verify that the subgroup is closed under group operations and that it contains the identity element to ensure it satisfies the subgroup criteria.
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Math Problem Analysis
Mathematical Concepts
Group Theory
Cosets
Finite Groups
Formulas
|G| = |G : H| * |H|
Theorems
Lagrange's Theorem
Suitable Grade Level
Undergraduate Mathematics
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