Question

Which of the following statements is/are TRUE ?

• A. The additive group of real numbers is isomorphic to the multiplicative group of positive real numbers
• B. The multiplicative group of nonzero real numbers is isomorphic to the multiplicative group of nonzero complex numbers
• C. The additive group of real numbers is isomorphic to the multiplicative group of nonzero complex numbers
• D. The additive group of real numbers is isomorphic to the additive group of rational numbers

Answer 1

The Correct Option is A

- (A) The additive group of real numbers \( (\mathbb{R}, +) \) is isomorphic to the multiplicative group of positive real numbers \( (\mathbb{R}^+, \cdot) \). This is true because the mapping \( f(x) = e^x \) is a bijective homomorphism between these two groups, with the inverse given by \( f^{-1}(x) = \log(x) \). Hence, these two groups are isomorphic. - (B) The multiplicative group of nonzero real numbers is not isomorphic to the multiplicative group of nonzero complex numbers. The structure of these two groups differs because \( \mathbb{R}^* \) (nonzero real numbers) is one-dimensional, while \( \mathbb{C}^* \) (nonzero complex numbers) is two-dimensional. Therefore, they are not isomorphic. - (C) The additive group of real numbers is not isomorphic to the multiplicative group of nonzero complex numbers. The additive group of real numbers is unbounded, whereas the multiplicative group of nonzero complex numbers is topologically different, and they do not share the same structure. - (D) The additive group of real numbers is isomorphic to the additive group of rational numbers. This is not true because \( \mathbb{R} \) is a continuous group, while \( \mathbb{Q} \) is discrete, so they cannot be isomorphic. Thus, the correct answer is (A).