Question:
A noncyclic group among the following is
A noncyclic group among the following is
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If every non-identity element in a group of order 4 has order 2, it is the non-cyclic Klein 4-group.
Updated On: Jul 3, 2026
- $(\{1, \omega, \omega^2\}, \cdot)$
- $(Z_4, +_4)$
- $(\{1, 3, 5, 7\}, \times_8)$
- $(Z, +)$
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The Correct Option is C
Solution and Explanation
Step 1: Concept
A group is cyclic if there exists an element $a$ such that every element can be written as $a^n$.
Step 2: Meaning
We test the powers of elements in each set under the given operation.
Step 3: Analysis
In $(\{1, 3, 5, 7\}, \times_8)$, the square of every element is 1: $3^2=9 \equiv 1$, $5^2=25 \equiv 1$, $7^2=49 \equiv 1$. No single element can generate the whole set $\{1, 3, 5, 7\}$.
Step 4: Conclusion
Since no generator exists, this group (known as the Klein 4-group $V_4$) is non-cyclic. Final Answer: (C)
A group is cyclic if there exists an element $a$ such that every element can be written as $a^n$.
Step 2: Meaning
We test the powers of elements in each set under the given operation.
Step 3: Analysis
In $(\{1, 3, 5, 7\}, \times_8)$, the square of every element is 1: $3^2=9 \equiv 1$, $5^2=25 \equiv 1$, $7^2=49 \equiv 1$. No single element can generate the whole set $\{1, 3, 5, 7\}$.
Step 4: Conclusion
Since no generator exists, this group (known as the Klein 4-group $V_4$) is non-cyclic. Final Answer: (C)
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