EXAMPLES OF GROUP HOMOMORPHISMS (1) Prove that (one line) GLn(R) → R× sending A ↦→ detA is a group homomorphism 1 Find its kernel (2)
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[PDF] 18703 Modern Algebra, Homomorphisms and kernels - MIT
Let φ: G -→ H be a group homomorphism The kernel of φ, denoted Ker φ, is the inverse image of the identity Then Ker φ is a subgroup of G Proof
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Prove that the kernel of a homomorphism is a subgroup of the domain of the homomorphism Let ϕ : G → G/ be a homomorphism, then you must show three things: 1 closure: Pick two arbitrary elements say a, b ∈ ker(ϕ) and show that their product must necessarily also be an element of ker(ϕ), i e show ϕ(ab) = 1
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EXAMPLES OF GROUP HOMOMORPHISMS (1) Prove that (one line) GLn(R) → R× sending A ↦→ detA is a group homomorphism 1 Find its kernel (2)
[PDF] (Group Homomorphism) - FacStaff Home Page for CBU
The kernel of a homomorphism φ : G → G is the set Kerφ = {x ∈ Gφ(x) = e} Example (1) Every isomorphism is a homomorphism with Kerφ = {e} (2) Let G = Z
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The kernel of a group homomorphism ϕ: G → H is a normal subgroup of G ϕ(g · k · g−1) = ϕ(g) · ϕ(k) · ϕ(g−1) = ϕ(g) · uH · ϕ(g)−1 = uH πG/N (g · h−1) = N · g · h−1 = N · g · N · h−1 = (N · g) · (N · h)−1, 1Note that since N ⊴ G, we have that N · g = g · N, for all g ∈ G
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In fact the opposite is true, every normal subgroup is the kernel of a homomorphism: Theorem 7 1 If H is a normal subgroup of a group G then the map γ: G −→ G/
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And, the kernel ker(ϕ) is a subgroup of G Proof Exercise The Trivial Homomorphisms: 1 Let G, G/ be groups Define ϕ : G
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The second subgroup if the kernel of ϕ, which is defined to be the set of all elements of G which get mapped to the identity element of H by ϕ: Ker (ϕ) = {g ∈ G : ϕ(
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7 1 Homomorphisms, Kernels and Images Definition 7 1 Let φ : G → L be a homomorphism of multiplicative groups The kernel and image of φ are the sets
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A subgroup K of a group G is normal if xKx-1 = K for all x ∈ G Let G and H be groups and let φ : G −→ H be a homomorphism Then the kernel ker(φ) of
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