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Let G be a group of order 2023. Which of the following statements are true?
(a.) G is an Abelian group
(b.) G is cyclic group
(c.) G is a simple group
(d.) G is not a simple group?
(a.) G is an Abelian group
(b.) G is cyclic group
(c.) G is a simple group
(d.) G is not a simple group?
Most Upvoted Answer
Let G be a group of order 2023. Which of the following statements are ...
Group Order and Structure
The order of group G is 2023, which can be factored as 2023 = 7 * 17 * 17. This indicates that G has a composition of prime factors.
Statement Analysis
Conclusion
In summary, the true statements about group G of order 2023 are only:
- (d) G is not a simple group.
The other statements (a), (b), and (c) are not necessarily true.
The order of group G is 2023, which can be factored as 2023 = 7 * 17 * 17. This indicates that G has a composition of prime factors.
Statement Analysis
- (a) G is an Abelian group: Not necessarily true. While groups of prime order are Abelian, G has a composite order. A group of order 2023 can have non-Abelian structures, especially since it can have normal subgroups.
- (b) G is a cyclic group: Not necessarily true. A cyclic group of order 2023 would imply that there's a single element generating the entire group. However, with the presence of multiple prime factors, G could be non-cyclic.
- (c) G is a simple group: False. A simple group has no normal subgroups other than the trivial group and itself. Given that 2023 has multiple prime factors, G will likely have normal subgroups corresponding to those factors.
- (d) G is not a simple group: True. Since G has order 2023 = 7 * 17 * 17, it must have normal subgroups, thus making it not simple.
Conclusion
In summary, the true statements about group G of order 2023 are only:
- (d) G is not a simple group.
The other statements (a), (b), and (c) are not necessarily true.
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Let G be a group of order 2023. Which of the following statements are true? (a.) G is an Abelian group (b.) G is cyclic group (c.) G is a simple group (d.) G is not a simple group?
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Let G be a group of order 2023. Which of the following statements are true? (a.) G is an Abelian group (b.) G is cyclic group (c.) G is a simple group (d.) G is not a simple group? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let G be a group of order 2023. Which of the following statements are true? (a.) G is an Abelian group (b.) G is cyclic group (c.) G is a simple group (d.) G is not a simple group? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let G be a group of order 2023. Which of the following statements are true? (a.) G is an Abelian group (b.) G is cyclic group (c.) G is a simple group (d.) G is not a simple group?.
Let G be a group of order 2023. Which of the following statements are true? (a.) G is an Abelian group (b.) G is cyclic group (c.) G is a simple group (d.) G is not a simple group? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let G be a group of order 2023. Which of the following statements are true? (a.) G is an Abelian group (b.) G is cyclic group (c.) G is a simple group (d.) G is not a simple group? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let G be a group of order 2023. Which of the following statements are true? (a.) G is an Abelian group (b.) G is cyclic group (c.) G is a simple group (d.) G is not a simple group?.
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