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

Let's analyze and justify each statement carefully. --- ### (i) \( \varphi(n) = n-1 \forall n \in \mathbb{N} \), where \( \varphi \) is the Euler-phi function. False. The Euler totient function \( \varphi(n) \) counts the number of integers \( k \) in \( \{1, 2, \dots, n\} \) that are coprime to \( n \). - If \( n = 2 \), then \( \varphi(2) = 1 \neq 2-1 \). - If \( n = 3 \), then \( \varphi(3) = 2 = 3-1 \), which holds. - However, for \( n = 4 \), \( \varphi(4) = 2 \neq 4-1 \). This fails in general. It holds only for prime numbers, where \( \varphi(p) = p-1 \). --- ### (ii) If \( G_1 \) and \( G_2 \) are groups, and \( f: G_1 \to G_2 \) is a group homomorphism, then \( 2 | G_1 | = 0 | G_2 | \). The notation appears unclear, but assuming it means \( 2 \cdot |G_1| = 0 \cdot |G_2| \), this is false in general. - If \( |G_1| \) is finite, multiplying by 2 gives a finite even number, which does not necessarily equal \( 0 \cdot |G_2| \). - If \( G_2 \) is infinite, __________ ____ ___ ______ _______.
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