Math Problem Statement
Help me find number 2
Solution
Let's solve problem number 2 step by step.
Problem 2(a): Show that defined by , , is a permutation on .
Solution:
- To prove that is a permutation, we need to show that is bijective (both injective and surjective):
- Injective:
Suppose .
This implies . Since and is a group, has an inverse . Multiplying both sides of the equation by , we get:
Therefore, is injective. - Surjective:
Let . We need to show that there exists such that , i.e., .
Since is in and is a group, the inverse exists. Multiply both sides of by :
Since , this shows is surjective.
- Injective:
Hence, is bijective and therefore a permutation on .
Problem 2(b): Show that is a subgroup of .
Solution:
- is the group of all permutations on .
- To prove is a subgroup of , we use the subgroup criterion:
- Closure under composition:
Let . Then for any ,
Since , . Thus, is closed under composition. - Identity element:
The identity map in is , where is the identity element of . Since , . - Inverses:
For , we need to show that . Note that for any ,
Since , .
- Closure under composition:
Therefore, is a subgroup of .
Problem 2(c): Show that the map defined by , is an isomorphism.
Solution:
- To prove is an isomorphism, we need to show that is a homomorphism, injective, and surjective:
- Homomorphism:
For ,
By definition of ,
so the inverse is . On the other hand,
and for any ,
Hence, , so is a homomorphism. - Injective:
Suppose . This implies , which means for all ,
[
- Homomorphism:
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Math Problem Analysis
Mathematical Concepts
Group Theory
Permutations
Subgroups
Isomorphisms
Formulas
ρ_a(x) = ax for all x ∈ G
Composition of functions: (ρ_a ∘ ρ_b)(x) = ρ_a(ρ_b(x))
Identity map: ρ_e(x) = x
Inverse map: ρ_a^{-1}(x) = a^{-1}x
Theorems
Cayley's Theorem
Group Isomorphism Properties
Subgroup Closure Criterion
Suitable Grade Level
Undergraduate (Advanced Mathematics)
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