Question:
Which one of the following is TRUE?
Which one of the following is TRUE?
Show Hint
A group \( \mathbb{Z}_n \) is cyclic if and only if \( n \) is a prime number, which means that every element of \( \mathbb{Z}_n \) is a power of some generator.
Updated On: Dec 15, 2025
- \( \mathbb{Z}_n \) is cyclic if and only if \( n \) is prime
- Every proper subgroup of \( \mathbb{Z}_n \) is cyclic
- Every proper subgroup of \( S_4 \) is cyclic
- If every proper subgroup of a group is cyclic, then the group is cyclic
Hide Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Check each option:
(A) False.
\(\mathbb Z_n\) is cyclic for all (n), not only when (n) is prime.
(B) True.
\(\mathbb Z_n\) is cyclic, and every subgroup of a cyclic group is cyclic. Hence every proper subgroup of \(\mathbb Z_n\) is cyclic.
(C) False.
\(S_4\) has proper subgroups that are not cyclic (e.g. \(V_4\)).
(D) False.
Counterexample: the Klein four group \(V_4\) is not cyclic, but all its proper subgroups are cyclic.
Correct answer: B
Was this answer helpful?
0
0
Top Questions on Group Theory
- Let \( G \) be a group with identity element \( e \), and let \( g, h \in G \) be such that the following hold: \[ g \neq e, \quad g^2 = e, \quad h \neq e, \quad h^2 \neq e, \quad {and} \quad ghg^{-1} = h^2. \] Then, the least positive integer \( n \) for which \( h^n = e \) is (in integer).
- Let \( G \) be a group with identity element \( e \), and let \( g, h \in G \) be such that the following hold: \[ g \neq e, \quad g^2 = e, \quad h \neq e, \quad h^2 \neq e, \quad {and} \quad ghg^{-1} = h^2. \] Then, the least positive integer \( n \) for which \( h^n = e \) is (in integer).
- Which of the following statements is/are TRUE ?
- IIT JAM MA - 2024
- Linear Algebra
- Group Theory
- The center Z(G) of a group G is defined as
z(G) = {x ∈ G ∶ xg = gx for all g ∈ G}.
Let |G| denote the order of G. Then, which of the following statements is/are TRUE for any group G ?- IIT JAM MA - 2024
- Linear Algebra
- Group Theory
- Consider
\(𝐺 = \left\{𝑚 + 𝑛\sqrt2\ ∶\ 𝑚, 𝑛 ∈ \Z\right\}\)
as a subgroup of the additive group ℝ.
Which of the following statements is/are TRUE ?- IIT JAM MA - 2024
- Linear Algebra
- Group Theory
View More Questions
Questions Asked in IIT JAM MA exam
- The sum of the infinite series \[ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} \pi^{2n+1}}{2^{2n+1} (2n)!} \] is equal to
- IIT JAM MA - 2025
- Sequences and Series of real numbers
- Let \( f(x) = 10x^2 + e^x - \sin(2x) - \cos x \), \( x \in \mathbb{R} \). The number of points at which the function \( f \) has a local minimum is:
- IIT JAM MA - 2025
- Mathematics
- Let \( f, g : \mathbb{R} \to \mathbb{R} \) be two functions defined by \[ f(x) = \begin{cases} x |x| \sin \frac{1}{x} & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \] and \[ g(x) = \begin{cases} x^2 \sin \frac{1}{x} + x \cos \frac{1}{x} & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \] Then, which one of the following is TRUE?
- IIT JAM MA - 2025
- Limit and Continuity
- Let \( T \) denote the triangle in the \( xy \)-plane bounded by the \( x \)-axis and the lines \( y = x \) and \( x = 1 \). The value of the double integral (over \( T \)) \[ \iint_T (5 - y) \, dx \, dy \] is equal to ............. (rounded off to two decimal places).
- IIT JAM MA - 2025
- Calculus
- Let \( f, g : \mathbb{R} \to \mathbb{R} \) be two functions defined by \[ f(x) = \begin{cases} |x|^{1/8} \sin \tfrac{1}{|x|} \cos x & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \] and \[ g(x) = \begin{cases} e^x \cos \tfrac{1}{x} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases} \] Then, which one of the following is TRUE?
- IIT JAM MA - 2025
- Limit and Continuity
View More Questions