Question:
Let H be a subgroup of a group G, where $O(H) = m$ and $O(G) = n$. Then, for any $a \in G$, $O(aH) = $
Let H be a subgroup of a group G, where $O(H) = m$ and $O(G) = n$. Then, for any $a \in G$, $O(aH) = $
Show Hint
All cosets of a subgroup $H$ have the same size as $H$ itself. They "partition" the group into equal-sized pieces.
Updated On: Jul 3, 2026
- $m$
- $n$
- $mn$
- $m+n$
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Concept
This relates to the properties of cosets in group theory.
Step 2: Meaning
$aH$ denotes a left coset of $H$ in $G$, which is the set $\{ah : h \in H\}$.
Step 3: Analysis
There is a one-to-one correspondence between $H$ and any of its cosets $aH$ defined by the map $f(h) = ah$.
Step 4: Conclusion
Because the map is a bijection, the number of elements (order) in the coset $aH$ is exactly the same as the order of the subgroup $H$, which is $m$. Final Answer: (A)
This relates to the properties of cosets in group theory.
Step 2: Meaning
$aH$ denotes a left coset of $H$ in $G$, which is the set $\{ah : h \in H\}$.
Step 3: Analysis
There is a one-to-one correspondence between $H$ and any of its cosets $aH$ defined by the map $f(h) = ah$.
Step 4: Conclusion
Because the map is a bijection, the number of elements (order) in the coset $aH$ is exactly the same as the order of the subgroup $H$, which is $m$. Final Answer: (A)
Was this answer helpful?
0
0
Top AP ECET BSc Mathematics Mathematics Questions
- The integrating factor of $x \cos x \frac{dy}{dx} + (x \sin x + \cos x) y = 1$ is}
- If $y = y(x)$ is the solution of $\left(\frac{2 + \sin x}{y + 1}\right) \frac{dy}{dx} + \cos x = 0$ with $y(0) = 1$, then $y\left(\frac{\pi}{2}\right) = $
- The solution of $y^{4}dx + 2xy^{3}dy = \frac{ydx - xdy}{x^{3}y^{3}}$ is}
- Which of the following is an integrating factor of $(x^{2} + y^{2} + 2x)dx + 2ydy = 0$?
- The differential equation $\frac{dy}{dx} = \frac{-(x + x^{8} + py^{2})}{y^{8} - y + qxy}$ is exact if}
View More Questions
Top AP ECET BSc Mathematics Algebra Questions
- On the set of integers Z, the operation defined as following is not a binary operation.
- In the group $(R-\{-1\}, *)$ where $*$ is defined by $a*b = a+b+ab, \forall a,b \in R-\{-1\}$, inverse of the element 2 is}
- The intersection of the subgroups 2Z, 3Z and 5Z, i.e., $2Z \cap 3Z \cap 5Z$ of (Z, +) is
- The permutations $f = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 4 & 8 & 2 & 5 & 1 & 3 & 7 & 6 \end{pmatrix}, g = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 5 & 1 & 2 & 8 & 7 & 4 & 3 & 6 \end{pmatrix}$ of $S_8$ are respectively}
- A noncyclic group among the following is
View More Questions
Top AP ECET BSc Mathematics Questions
- The integrating factor of $x \cos x \frac{dy}{dx} + (x \sin x + \cos x) y = 1$ is}
- If $y = y(x)$ is the solution of $\left(\frac{2 + \sin x}{y + 1}\right) \frac{dy}{dx} + \cos x = 0$ with $y(0) = 1$, then $y\left(\frac{\pi}{2}\right) = $
- The solution of $y^{4}dx + 2xy^{3}dy = \frac{ydx - xdy}{x^{3}y^{3}}$ is}
- Which of the following is an integrating factor of $(x^{2} + y^{2} + 2x)dx + 2ydy = 0$?
- The differential equation $\frac{dy}{dx} = \frac{-(x + x^{8} + py^{2})}{y^{8} - y + qxy}$ is exact if}
View More Questions