Question:
Let \( C_n \) denote a cyclic group having \( n \) elements. If there is a surjective group homomorphism from \( C_n \) to \( C_{30} \), then the total number of such distinct surjective homomorphisms is .
Let \( C_n \) denote a cyclic group having \( n \) elements. If there is a surjective group homomorphism from \( C_n \) to \( C_{30} \), then the total number of such distinct surjective homomorphisms is .
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The number of surjective homomorphisms from a cyclic group onto \(C_m\) is equal to the number of generators of \(C_m\), which is \(\phi(m)\), whenever such a map exists.
Updated On: Jun 1, 2026
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Correct Answer: 8
Solution and Explanation
Step 1: Use property of cyclic groups.
A homomorphism from a cyclic group is completely determined by the image of its generator.
Step 2: Let \(g\) be a generator of \(C_n\).
If \( \phi:C_n\to C_{30} \) is surjective, then \( \phi(g) \) must generate \(C_{30}\).
Step 3: Count generators of \(C_{30}\).
The number of generators of a cyclic group of order \(30\) is
\[ \phi(30) \]
Step 4: Evaluate Euler phi function.
\[ 30=2\cdot 3\cdot 5 \]
\[ \phi(30)=30\left(1-\frac12\right)\left(1-\frac13\right)\left(1-\frac15\right) \]
Step 5: Simplify.
\[ \phi(30)=30\cdot \frac12\cdot \frac23\cdot \frac45 \]
\[ \phi(30)=8 \]
Step 6: Final conclusion.
Hence, the total number of distinct surjective homomorphisms is
\[ \boxed{8} \]
A homomorphism from a cyclic group is completely determined by the image of its generator.
Step 2: Let \(g\) be a generator of \(C_n\).
If \( \phi:C_n\to C_{30} \) is surjective, then \( \phi(g) \) must generate \(C_{30}\).
Step 3: Count generators of \(C_{30}\).
The number of generators of a cyclic group of order \(30\) is
\[ \phi(30) \]
Step 4: Evaluate Euler phi function.
\[ 30=2\cdot 3\cdot 5 \]
\[ \phi(30)=30\left(1-\frac12\right)\left(1-\frac13\right)\left(1-\frac15\right) \]
Step 5: Simplify.
\[ \phi(30)=30\cdot \frac12\cdot \frac23\cdot \frac45 \]
\[ \phi(30)=8 \]
Step 6: Final conclusion.
Hence, the total number of distinct surjective homomorphisms is
\[ \boxed{8} \]
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