Question

How many elements of order 5 are there in a cyclic group of order 25?

• A. \(1\)
• B. \(4\)
• C. \(5\)
• D. \(10\)

Hint:

In cyclic groups: Number of elements of order \(d\) = \( \phi(d) \)

Answer 1

The Correct Option is B

Concept: Order of elements in a cyclic group
In a cyclic group of order \( n \), the number of elements of order \( d \) (where \( d \mid n \)) is given by Euler’s totient function: \[ \phi(d) \]
Step 1: Identify the group order
\[ n = 25 = 5^2 \]
Step 2: Find valid divisors
Possible orders divide 25: \[ 1, 5, 25 \] We need elements of order \(5\).
Step 3: Apply Euler's totient function
\[ \phi(5) = 5 - 1 = 4 \]
Step 4: Interpretation
There are exactly 4 generators of the subgroup of order 5. Conclusion: \[ \text{Number of elements of order 5} = 4 \]