can you help me with a,b please. abstarct algebra permutations 5. Let G be a group and let G act on itself by conjugation. For a E G, denote Oa the orbit of aunder conjugation action. The orbits under the conjugation action are called the conjugacyclasses of G.(a) Find the kernel of the induced permutation representation p: G→ SG.(b) Show that the action by conjugation is faithful if and only if the center of G is trivial.i.e.: Z(G) = {1}

Name: can you help me with a,b please. abstarct algebra permutations 5. Let
Uploaded: 2024-03-05T11:53:49.000Z
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Description: can you help me with a,b please. abstarct algebra permutations 5. Let G be a group and let G act on itself by conjugation. For a E G, denote Oa the orbit of aunder conjugation action. The orbits under the conjugation action are called the conjugacycl

Let G be a group and let G act on itself by conjugation. For a∈G, Use 𝒪a to denote the orbit of a under conjugation action. 𝒪a- Conjugacy classes of G. (a) To find the kernel of the induced permutation ρ:G→SG. Kernel of the group, K=a∈G|a.g=g ∀g∈G The permutation representation is a group homomorphism ρ:G→SG is given by ρag=a.g and ker ρ=a∈G|ρ(a)=1k⊆ker ρ, for k∈K then ∀g∈G,ρkg=k.g=g⇒ρk=idG=1 Hence for, a∈ker ρ ker ρ⊆k, let k∈ρ kerρ.∀ g∈G⇒k.a=ρkg=idGg=a Hence k∈K. Thus, the kernel of the induced permutation is K=a∈G|a.g=g ∀g∈G(b)To show that the action by the conjugation is faithful iff the center of G is trivial: i.e., ZG=1. If G is faithful, then For g1≠g2 in G, there is an x∈X ∋ g1.x≠g2.x Now, g1.x=g2.x⇒g1=g2,∀g∈G Center of the conjugation group G, Z(G)=x∈G|x=g2-1g1=1 ∀g∈G,g1.x=g2.x⇔g1gg1-1=g2gg2-1 ⇔g2-1g1gg1-1=g2-1g2gg2-1 ⇔g2-1g1gg1-1=gg2-1 ⇔g2-1gg1-1=gg2-1g1x=g2-1g1⇔gg2-1g1=g2-1g1g Which is the center of G. Thus, the action by conjugation is faithful iff the center of G is trivial.