Question

Consider the group $G = \{+1, -1, +i, -i\}$ under multiplication operation. Which of the following is not correct?

• A. $G$ is an abelian group
• B. $G$ is a cyclic group
• C. $\{1, -1\}$ is a subgroup of $G$
• D. $\{1, -i, i\}$ is a subgroup of $G$

Hint:

Lagrange's Theorem is your best friend for group theory MCQs. If the size of the "subgroup" doesn't divide the size of the main group, it's an impostor!

Answer 1

The Correct Option is D

Let's analyze the properties of the group of fourth roots of unity, $G = \{1, -1, i, -i\}$. 1. Group Classification:
• Abelian: Multiplication of complex numbers is commutative ($ab = ba$). Thus, $G$ is abelian.
• Cyclic: A group is cyclic if it can be generated by a single element. In $G$, the element $i$ is a generator because: $i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1$. Since $i$ generates all elements, $G$ is cyclic. 2. Subgroup Requirements: According to Lagrange's Theorem, the order (size) of a subgroup must divide the order of the group. The order of $G$ is 4.
• Option (3): $\{1, -1\}$ has order 2. Since 2 divides 4 and it satisfies group axioms, it is a valid subgroup.
• Option (4): $\{1, -i, i\}$ has order 3. Since 3 does not divide 4, it cannot be a subgroup of $G$. Furthermore, it is not closed under multiplication (e.g., $i \times i = -1$, which is not in the set). 3. Conclusion: Statement (4) is incorrect because the set does not satisfy the closure property and its size violates Lagrange's theorem.