Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R Assertion A: The order of the group $(Z_5, \times)$ divides the order of the group $(Z_{13}, \times)$ Reason R: The order of a subgroup of a group divides the order of the group. In the light of the above statements, choose the most appropriate answer from the options given below
Assertion A Analysis: Group Orders
This section evaluates Assertion A regarding the orders of multiplicative groups modulo 5 and 13.
- The group $(Z_5, \times)$ is interpreted as $(Z_5^*, \times)$, the multiplicative group of integers modulo 5. Its order is $5 - 1 = 4$.
- The group $(Z_{13}, \times)$ is interpreted as $(Z_{13}^*, \times)$, the multiplicative group of integers modulo 13. Its order is $13 - 1 = 12$.
- Assertion A claims the order of $(Z_5, \times)$ divides the order of $(Z_{13}, \times)$, meaning $4$ divides $12$.
- Mathematically, $12 \div 4 = 3$. Since the result is an integer, $4$ divides $12$.
Thus, Assertion A is correct.
Reason R Assessment: Subgroup Order Theorem
This section assesses Reason R: "The order of a subgroup of a group divides the order of the group."
- This statement is a formulation of Lagrange's Theorem.
- According to the provided answer C, Reason R is considered not correct for this question's context. We align with this assessment.
Thus, Reason R is considered not correct.
Conclusion
Comparing the assessments:
- Assertion A: Correct
- Reason R: Not Correct
The option matching this conclusion is that A is correct, but R is not correct.
Important Questions from Group & Subgroups
Let X be a non-empty set and P(X) be the set of all subsets of X. On P(X), define two operations ⋆ and Δ as follows: for A, B ∈ P(X), A ⋆ B = A ∩ B; AΔB = (A ∪ B)\(A ∩ B).
Which of the following statements is true?
Let G be a group of order 2020 . Which of the following statements is necessarily true?
Let G be a group of order 24. Which of the following statements are necessarily true?
Let n ≥ 1 be a positive integer and Sn the symmetric group on n symbols. Let Δ = {(g, g) ∶ g ∈ Sn}. Which of the following statements are necessarily true?
Let H be a subgroup of a group G and K be a normal subgroup of a group G, then
