GATE Exam > GATE Questions > Let (G, *) be a group such that O(G) = 8, wh...
Start Learning for Free
Let (G, *) be a group such that O(G) = 8, where O(G) denotes the order of the group G. Which of the following is true?
- a)There exists no element a in G such that O(A) = 6.
- b)There exists an element in G such that O(a)= 4.
- c)There exists more than one element a in G such that O(A) = 1
- d)None of these
Correct answer is option 'D'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
Let (G, *) be a group such that O(G) = 8, where O(G) denotes the orde...
Theorem: If a is an art element of G, then. O(A) must divide O(G).
View all questions of this test
Therefore (A) is false, as O(A) = 6 and 6 doesn’t divide 8.
However the converse of the above theorem need not be true, therefore (B) is false.
(C) says there are at least two elements in G with order 1, however this violates the fact that every group can have only one identity element. So (C) is also wrong. So the appropriate choice is (d).
Most Upvoted Answer
Let (G, *) be a group such that O(G) = 8, where O(G) denotes the orde...
Explanation:
Given:
- Let (G, *) be a group with O(G) = 8.
Analysis:
- The order of an element 'a' in a group G, denoted by O(a), is defined as the smallest positive integer 'n' such that a^n = e, where e is the identity element of the group.
Option a):
- There could exist an element 'a' in G such that O(a) = 6. It is not contradictory to the given information about the order of the group.
Option b):
- There could exist an element 'a' in G such that O(a) = 4. This is also consistent with the given information about the order of the group.
Option c):
- The order of any element in a group is always greater than or equal to 1. So, there cannot be more than one element with O(a) = 1 in the group.
Conclusion:
- Since both options a) and b) are possible based on the given information, the correct answer is option 'D' (None of these).
Given:
- Let (G, *) be a group with O(G) = 8.
Analysis:
- The order of an element 'a' in a group G, denoted by O(a), is defined as the smallest positive integer 'n' such that a^n = e, where e is the identity element of the group.
Option a):
- There could exist an element 'a' in G such that O(a) = 6. It is not contradictory to the given information about the order of the group.
Option b):
- There could exist an element 'a' in G such that O(a) = 4. This is also consistent with the given information about the order of the group.
Option c):
- The order of any element in a group is always greater than or equal to 1. So, there cannot be more than one element with O(a) = 1 in the group.
Conclusion:
- Since both options a) and b) are possible based on the given information, the correct answer is option 'D' (None of these).
|
Explore Courses for GATE exam
|
|
Let (G, *) be a group such that O(G) = 8, where O(G) denotes the order of the group G. Which of the following is true?a)There exists no element a in G such that O(A) = 6.b)There exists an element in G such that O(a)= 4.c)There exists more than one element a in G such that O(A) = 1d)None of theseCorrect answer is option 'D'. Can you explain this answer?
Question Description
Let (G, *) be a group such that O(G) = 8, where O(G) denotes the order of the group G. Which of the following is true?a)There exists no element a in G such that O(A) = 6.b)There exists an element in G such that O(a)= 4.c)There exists more than one element a in G such that O(A) = 1d)None of theseCorrect answer is option 'D'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Let (G, *) be a group such that O(G) = 8, where O(G) denotes the order of the group G. Which of the following is true?a)There exists no element a in G such that O(A) = 6.b)There exists an element in G such that O(a)= 4.c)There exists more than one element a in G such that O(A) = 1d)None of theseCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let (G, *) be a group such that O(G) = 8, where O(G) denotes the order of the group G. Which of the following is true?a)There exists no element a in G such that O(A) = 6.b)There exists an element in G such that O(a)= 4.c)There exists more than one element a in G such that O(A) = 1d)None of theseCorrect answer is option 'D'. Can you explain this answer?.
Let (G, *) be a group such that O(G) = 8, where O(G) denotes the order of the group G. Which of the following is true?a)There exists no element a in G such that O(A) = 6.b)There exists an element in G such that O(a)= 4.c)There exists more than one element a in G such that O(A) = 1d)None of theseCorrect answer is option 'D'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Let (G, *) be a group such that O(G) = 8, where O(G) denotes the order of the group G. Which of the following is true?a)There exists no element a in G such that O(A) = 6.b)There exists an element in G such that O(a)= 4.c)There exists more than one element a in G such that O(A) = 1d)None of theseCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let (G, *) be a group such that O(G) = 8, where O(G) denotes the order of the group G. Which of the following is true?a)There exists no element a in G such that O(A) = 6.b)There exists an element in G such that O(a)= 4.c)There exists more than one element a in G such that O(A) = 1d)None of theseCorrect answer is option 'D'. Can you explain this answer?.
Solutions for Let (G, *) be a group such that O(G) = 8, where O(G) denotes the order of the group G. Which of the following is true?a)There exists no element a in G such that O(A) = 6.b)There exists an element in G such that O(a)= 4.c)There exists more than one element a in G such that O(A) = 1d)None of theseCorrect answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for GATE.
Download more important topics, notes, lectures and mock test series for GATE Exam by signing up for free.
Here you can find the meaning of Let (G, *) be a group such that O(G) = 8, where O(G) denotes the order of the group G. Which of the following is true?a)There exists no element a in G such that O(A) = 6.b)There exists an element in G such that O(a)= 4.c)There exists more than one element a in G such that O(A) = 1d)None of theseCorrect answer is option 'D'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
Let (G, *) be a group such that O(G) = 8, where O(G) denotes the order of the group G. Which of the following is true?a)There exists no element a in G such that O(A) = 6.b)There exists an element in G such that O(a)= 4.c)There exists more than one element a in G such that O(A) = 1d)None of theseCorrect answer is option 'D'. Can you explain this answer?, a detailed solution for Let (G, *) be a group such that O(G) = 8, where O(G) denotes the order of the group G. Which of the following is true?a)There exists no element a in G such that O(A) = 6.b)There exists an element in G such that O(a)= 4.c)There exists more than one element a in G such that O(A) = 1d)None of theseCorrect answer is option 'D'. Can you explain this answer? has been provided alongside types of Let (G, *) be a group such that O(G) = 8, where O(G) denotes the order of the group G. Which of the following is true?a)There exists no element a in G such that O(A) = 6.b)There exists an element in G such that O(a)= 4.c)There exists more than one element a in G such that O(A) = 1d)None of theseCorrect answer is option 'D'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Let (G, *) be a group such that O(G) = 8, where O(G) denotes the order of the group G. Which of the following is true?a)There exists no element a in G such that O(A) = 6.b)There exists an element in G such that O(a)= 4.c)There exists more than one element a in G such that O(A) = 1d)None of theseCorrect answer is option 'D'. Can you explain this answer? tests, examples and also practice GATE tests.
|
|
Explore Courses for GATE exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup