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) $\phi(n)=n-1\forall n\in \mathbb{N},$ where $\phi$ is the Euler-phi function.

ii) If $G_{1}$ and $G_{2}$ are groups, and $f:G_{1}\rightarrow G_2$ is a group homomorphism, then $o(G_1)=o(G_2).$

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

iv) If G is a group and $H\underline{\Delta}G$ ,then $\mid$ $G:H\mid=2.$

v) Every element of $S_n$ has order at most n .

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

vii) If $\sigma \in S_n\left ( n\geq 3 \right )$ is a product of an even number of disjoint cycles, then sign $\left ( \sigma \right )=1.$

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 $\left [ 0,1 \right ]$ to $\mathbb{R}$ form a ring with respect to pointwise addition and multiplication.

11 Mar 2024

Answer :

Word Count : 415

i) False. The Euler-phi function, denoted as φ(n), represents the count of positive integers less than n that are coprime to n. However, φ(n) = n - 1 only if n is a prime number. For example, φ(4) = 2 ≠ 4 - 1.

ii) False. The order of a group, denoted as |G|, represents the number of elements in the group. While a group homomorphism preserves the structure of groups, it does not necessarily preserve the number of elements. For instance, consider __________ __________ _________ ___ __________ __________ __________ ________ _______ _______ _____.
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