Question:
Consider the group \( (\mathbb{Q}, +) \) and its subgroup \( (\mathbb{Z}, +) \).
For the quotient group \( \mathbb{Q}/\mathbb{Z} \), which one of the following is FALSE?
Consider the group \( (\mathbb{Q}, +) \) and its subgroup \( (\mathbb{Z}, +) \).
For the quotient group \( \mathbb{Q}/\mathbb{Z} \), which one of the following is FALSE?
For the quotient group \( \mathbb{Q}/\mathbb{Z} \), which one of the following is FALSE?
Updated On: Oct 1, 2024
- \( \mathbb{Q}/\mathbb{Z} \) contains a subgroup isomorphic to \( (\mathbb{Z}, +) \).
- There is exactly one group homomorphism from \( \mathbb{Q}/\mathbb{Z} \) to \( (\mathbb{Q}, +) \).
- For all \( n \in \mathbb{N} \), there exists \( g \in \mathbb{Q}/\mathbb{Z} \) such that the order of \( g \) is \( n \).
- \( \mathbb{Q}/\mathbb{Z} \) is not a cyclic group.
Hide Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
The correct option is (A): \( \mathbb{Q}/\mathbb{Z} \) contains a subgroup isomorphic to \( (\mathbb{Z}, +) \).
Was this answer helpful?
0
0
Top Questions on Groups
- Which of the following statements is/are TRUE ?
- 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 ? - 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 ? - Let G be a group of order 39 such that it has exactly one subgroup of order 3 and exactly one subgroup of order 13. Then, which one of the following statements is TRUE ?
- Let A ⊆ \(\Z\) with 0 ∈ A. For r, s ∈ \(\Z\), define
rA = {ra : a ∈ A}, rA + sA = {ra + sb : a, b ∈ A}.
Which of the following conditions imply that A is a subgroup of the additive group \(\Z\) ?
View More Questions
Questions Asked in IIT JAM MA exam
- Consider the function f: ℝ → ℝ given by f(x) = x3 - 4x2 + 4x - 6.
For c ∈ ℝ, let
\(S(c)=\left\{x \in \R : f(x)=c\right\}\)
and |S(c)| denote the number of elements in S(c). Then, the value of
|S(-7)| + |S(-5)| + |S(3)|
equals _________- IIT JAM MA - 2024
- Functions of One Real Variable
- Let y : ℝ → ℝ be the solution to the differential equation
\(\frac{d^2y}{dx^2}+2\frac{dy}{dx}+5y=1\)
satisfying y(0) = 0 and y'(0) = 1.
Then, \(\lim\limits_{x \rightarrow \infin}y(x)\) equals __________ (rounded off to two decimal places).- IIT JAM MA - 2024
- Differential Equations
- For a > b > 0, consider
\(D=\left\{(x,y,z) \in \R^3 :x^2+y^2+z^2 \le a^2\ \text{and } x^2+y^2 \ge b^2\right\}.\)
Then, the surface area of the boundary of the solid D is- IIT JAM MA - 2024
- Functions of Two or Three Real Variables
- Let P7(x) be the real vector space of polynomials, in the variable x with real coefficients and having degree at most 7, together with the zero polynomial.Let T : P7(x) → P7(𝑥) be the linear transformation defined by
\(T(f(x))=f(x)+\frac{df(x)}{dx}\).
Then, which one of the following is TRUE ?- IIT JAM MA - 2024
- Finite Dimensional Vector Spaces
- For a positive integer \( n \), let \( U(n) = \{ r \in \mathbb{Z}_n : \gcd(r, n) = 1 \} \) be the group under multiplication modulo \( n \). Then, which one of the following statements is TRUE?
- IIT JAM MA - 2024
- Functions of One Real Variable
View More Questions