IGNOU MTE 6 2026 - Which of the following statements are true? Justify your ans

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.) (20)
i) $\phi(n) = n - 1 \,\, \forall n \in \mathbb{N}$ , where $\phi$ is the Euler-phi function.
ii) If G₁ and G₂ 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 $H \trianglelefteq G$ , then $|G : H| = 2$ .
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 $xr = rx \,\, \forall x \in I$ and $r \in R$ .
vii) If $\sigma \in S_n \,\, (n \ge 3)$ is a product of an even number of disjoint cycles, then $\text{sign }(\sigma) = 1$ .
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 $\mathbb{R}$ form a ring with respect to point-wise addition and multiplication.

09 Jan 2026

Answer :

Word Count : 342

i) False – The Euler-phi function (\phi(n)) counts numbers coprime to (n). There is no general equation given; without specifying the equation, this cannot be true in general. For example, (\phi(6) = 2) and (\phi(8) = 4); any arbitrary equation involving (\phi(n)) may fail. ii) True – If (f: G_1 \to G_2) is a group homomorphism, then (f(e_{G_1}) = e_{G_2}). This follows from the homomorphism property (f(e_{G_1}) = f(e_{G_1} * e_{G_1}) = f(e_{G_1}) f(e_{G_1})), so (f(e_{G_1})) must be the identity in (G_2). iii) False – Not

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