Number of elements of order p in Zp2qwhere p and q are distinct prime is;a)pb)p-1c)p2-1d)None of theseCorrect answer is option 'B'. Can you explain this answer?

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Answer

Understanding the Structure of Zp2q
The group Zp2q, where p and q are distinct primes, refers to the additive group of integers modulo p2q. To find the number of elements of order p, we need to explore the structure of this group.
Group Order
- The order of Zp2q is p2q.
- The total number of elements is p2q.
Element Order in Groups
- An element's order is the smallest positive integer n such that n * g = 0 (where g is the element).
- The order of an element divides the order of the group.
Finding Elements of Order p
- We seek elements g in Zp2q such that their order is p.
- The subgroup of order p is cyclic and consists of elements that, when multiplied by p, yield 0.
Counting Elements of Order p
- The number of elements of order p can be derived from the structure of the cyclic subgroup.
- The cyclic subgroup of order p has p-1 elements of order p (since all non-zero elements in a cyclic group of prime order are of full order).
Conclusion
- Therefore, the number of elements of order p in Zp2q is p-1.
- Hence, the correct answer is option 'B': p-1.
This analysis provides a clear understanding of why the number of elements of order p in Zp2q is p-1, based on the properties of cyclic groups and element orders.

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