Q1. Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so.)
- (i)) φ(n) = n-1 ∀ n ∈ N, where φ is the Euler-phi function. (83 words)
- (ii)) If G₁ and G₂ are groups, and f : G₁ → G₂ is a group homomorphism, then o(G₁) = o(G₂). (83 words)
- (iii)) If G is an abelian group, then G is cyclic. (83 words)
- (iv)) If G is a group and H ⊲ G, then | G : H | = 2. (83 words)
- (v)) Every element of Sn has order at most n. (83 words)
- (vi)) If R is a ring and I is an ideal of R, then xr = rx ∀ x ∈ I and r ∈ R. (83 words)
- (vii)) If σ ∈ Sn (n ≥ 3) is a product of an even number of disjoint cycles, then sign (σ) = 1. (83 words)
- (viii)) If a ring has a unit, then it has only one unit. (83 words)
- (ix)) The characteristic of a finite field is zero. (83 words)
- (x)) The set of discontinuous functions from [0, 1] to R form a ring with respect to point-wise addition and multiplication. (83 words)
Key Points:
- Euler-phi function φ(n)=n-1 only for prime n.
- Homomorphisms do not guarantee equal group orders (e.g., Z₄ to Z₂).
- Abelian groups are not necessarily cyclic (e.g., Klein four-group).
- Normal subgroup index |G:H| is not always 2 (e.g., Z₁₂ with H={0,3,6,9}).
Answer: This response evaluates ten statements related to Abstract Algebra, covering groups, rings, and fields, as typically taught in MTE-06. Each statement is analyzed for its truthfulness, and a justification or counterexample is provided based on fundamental definitions and theorems from the course. The aim is to demonstrate a comprehensive understanding of core algebraic concepts.