Flashcards on Group Theory: Core Concepts
Group Theory: Core Concepts Explained for Students
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Group Theory (Homomorphism and Isomorphism)
12 cards
Card 1
Question: What properties does a group homomorphism θ:G→H preserve for the identity and inverses?
Answer: θ(e_G)=e_H and θ(a^{-1})=θ(a)^{-1} for all a in G.
Card 2
Question: How does a homomorphism θ:G→H behave on integer powers of an element?
Answer: θ(a^n)=θ(a)^n for all integers n.
Card 3
Question: What can be said about the image θ(G) of a homomorphism θ:G→H?
Answer: θ(G) is a subgroup of H.
Card 4
Question: If a homomorphism θ:G→H is injective, what is the relationship between G and θ(G)?
Answer: G is isomorphic to θ(G) (G ≅ θ(G)).
Card 5
Question: Give an explicit isomorphism between Z_3 and the subgroup generated by the 3-cycle (123).
Answer: Define θ:Z_3→⟨(123)⟩ by θ([0])=e, θ([1])=(123), θ([2])=(132). This is a bijective homomorphism, so Z_3 ≅ ⟨(123)⟩.
Card 6
Question: Provide an isomorphism between Z (under addition) and the subgroup 2Z.
Answer: α:Z→2Z given by α(n)=2n is an isomorphism (homomorphism and bijective onto 2Z).
Card 7
Question: Provide an isomorphism between R_{>0} (under multiplication) and R (under addition).
Answer: γ:R_{>0}→R defined by γ(x)=ln x; since ln(xy)=ln x+ln y, γ is a homomorphism and bijection, so R_{>0} ≅ R.
Card 8
Question: What group properties are preserved by isomorphism?
Answer: Being abelian, being cyclic, element orders, and subgroup structure sizes are preserved by isomorphism.
Card 9
Question: State Cayley’s Theorem.
Answer: Every group G is isomorphic to a subgroup of the symmetric group on G; i.e. G ≅ a subgroup of Sym(G). If |G|=n then G ≅ a subgroup of S_n.
Card 10
Question: What map is used to prove Cayley’s Theorem and why is it a homomorphism and injective?
Answer: For each a∈G define λ_a:G→G by λ_a(x)=ax (left multiplication). The map θ:G→Sym(G) with θ(a)=λ_a is a homomorphism (θ(ab)=λ_{ab}=λ_a∘λ_b) and is injec