QB_PH_003: Linear Algebra & Group Theory

Let $G$ be a finite cyclic group generated by an element $g$ of order 12$. What is the order of the subgroup generated by $g^6$?

Correct Answer

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Governing Concept & Domain Context:** * This question evaluates **Fundamentals of Complex analysis** under *Complex analysis* in GATE PH. 2. **Stage 2: Mathematical Derivation:** * In group theory, for an element of order $n$, the order of power $g^k$ is $|g^k| = \frac{n}{\gcd(n, k)}$. Here $n = 12$ and $k = 6$. Since $\gcd(12, 6) = 6$, we obtain $|g^6| = \frac{12}{6} = 2$. 3. **Stage 3: Distractor Elimination Forensics:** * The evaluated correct answer is derived using standard physical equations. * Distractor values correspond to typical unit conversion omissions or inverted formulas. 4. **Stage 4: Common Student Trap:** * *Trap:* Multiplying orders instead of dividing by the greatest common divisor. *Reference: George B. Arfken, Hans J. Weber, Mathematical Methods for Physicists (Academic Press).*