Question

Consider the following statements:
A. Every finite cyclic group of order n is isomorphic to the group $(Z_n, +_n)$.
B. Every infinite cyclic group is isomorphic to the group (Z, +).
Which of the following statement is TRUE?

• A. both A and B are true
• B. A is true, B is false
• C. A is false, B is true
• D. both A and B are false

Hint:

Cyclic groups are the "simplest" groups—there is essentially only one for each possible size ($n$ or infinite).

Answer 1

The Correct Option is A

Step 1: Concept
This is a fundamental theorem in group theory regarding the classification of cyclic groups.

Step 2: Meaning
Isomorphism means two groups have the same structure despite having different elements.

Step 3: Analysis
Statement A: A finite cyclic group $\langle a \rangle$ of order $n$ maps perfectly to $\{0, 1, ..., n-1\}$ under $a^k \to k \pmod n$. Statement B: An infinite cyclic group $\langle a \rangle$ maps to $Z$ via $a^k \to k$.

Step 4: Conclusion
Both statements are foundational mathematical truths; thus, both A and B are true. Final Answer: (A)