Which of the following statements are true? Justify your answers. (Thi

Question

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) (n) = n-1∀n ∈ N, where o is the Euler-phi function.

ii) If G₁ and G₂ are groups, and f: G₁→ G₂ is a group homomorphism, then 2 o(G₁) = 0(G2).

If G is an abelian group, then G is cyclic.

iv) If G is a group and HAG, then | G: H=2.

v) Every element of S has order at most n.

vi) If R is a ring and I is an ideal of R, then xr = rx ∀ x ∈ I and re R.

vii) If σε S_n(n ≥3) is a product of an even number of disjoint cycles, then sign (σ)=1.

viii) If a ring has a unit, then it has only one unit.

ix) The characteristic of a finite field is zero.

x) The set of discontinuous functions from [0, 1] to R form a ring with respect to point- wise addition and multiplication.

18 Feb 2025

Answer :

Word Count : 843

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