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Which one of the following statements is true?
(a) Every group of order 12 has a non-trivial proper normal subgroup.
(b) Some group of order 12 does not have a non-trivial proper normal subgroup.
(c) Every group of order 12 has a subgroup of order 6.
(d) Every group of order 12 has an element of order 12.?
(a) Every group of order 12 has a non-trivial proper normal subgroup.
(b) Some group of order 12 does not have a non-trivial proper normal subgroup.
(c) Every group of order 12 has a subgroup of order 6.
(d) Every group of order 12 has an element of order 12.?
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Which one of the following statements is true?(a) Every group of order...
Analysis of Statements
To determine which statement is true regarding groups of order 12, we analyze each option based on group theory principles.
Statement (a): Every group of order 12 has a non-trivial proper normal subgroup.
- By the Sylow theorems, the number of Sylow p-subgroups must divide the group order and also satisfy certain congruences.
- For a group of order 12, it has Sylow 3-subgroups and Sylow 2-subgroups.
- Therefore, it can be shown that every group of order 12 has a non-trivial normal subgroup.
Statement (b): Some group of order 12 does not have a non-trivial proper normal subgroup.
- This statement contradicts (a). Since it has been established that every group of order 12 has a non-trivial proper normal subgroup, this statement is false.
Statement (c): Every group of order 12 has a subgroup of order 6.
- By Lagrange's theorem, the possible orders of subgroups divide the group order.
- There are groups of order 12 that do not have a subgroup of order 6, such as the alternating group A4, which has subgroups of orders 1, 2, and 3, but not 6.
Statement (d): Every group of order 12 has an element of order 12.
- According to the definition of group order, no element can have an order greater than the group order. Hence, no element can have an order equal to 12 in a group of order 12.
Conclusion
The true statement is:
- (a) Every group of order 12 has a non-trivial proper normal subgroup.
To determine which statement is true regarding groups of order 12, we analyze each option based on group theory principles.
Statement (a): Every group of order 12 has a non-trivial proper normal subgroup.
- By the Sylow theorems, the number of Sylow p-subgroups must divide the group order and also satisfy certain congruences.
- For a group of order 12, it has Sylow 3-subgroups and Sylow 2-subgroups.
- Therefore, it can be shown that every group of order 12 has a non-trivial normal subgroup.
Statement (b): Some group of order 12 does not have a non-trivial proper normal subgroup.
- This statement contradicts (a). Since it has been established that every group of order 12 has a non-trivial proper normal subgroup, this statement is false.
Statement (c): Every group of order 12 has a subgroup of order 6.
- By Lagrange's theorem, the possible orders of subgroups divide the group order.
- There are groups of order 12 that do not have a subgroup of order 6, such as the alternating group A4, which has subgroups of orders 1, 2, and 3, but not 6.
Statement (d): Every group of order 12 has an element of order 12.
- According to the definition of group order, no element can have an order greater than the group order. Hence, no element can have an order equal to 12 in a group of order 12.
Conclusion
The true statement is:
- (a) Every group of order 12 has a non-trivial proper normal subgroup.
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Which one of the following statements is true?(a) Every group of order 12 has a non-trivial proper normal subgroup.(b) Some group of order 12 does not have a non-trivial proper normal subgroup.(c) Every group of order 12 has a subgroup of order 6.(d) Every group of order 12 has an element of order 12.?
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Which one of the following statements is true?(a) Every group of order 12 has a non-trivial proper normal subgroup.(b) Some group of order 12 does not have a non-trivial proper normal subgroup.(c) Every group of order 12 has a subgroup of order 6.(d) Every group of order 12 has an element of order 12.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Which one of the following statements is true?(a) Every group of order 12 has a non-trivial proper normal subgroup.(b) Some group of order 12 does not have a non-trivial proper normal subgroup.(c) Every group of order 12 has a subgroup of order 6.(d) Every group of order 12 has an element of order 12.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which one of the following statements is true?(a) Every group of order 12 has a non-trivial proper normal subgroup.(b) Some group of order 12 does not have a non-trivial proper normal subgroup.(c) Every group of order 12 has a subgroup of order 6.(d) Every group of order 12 has an element of order 12.?.
Which one of the following statements is true?(a) Every group of order 12 has a non-trivial proper normal subgroup.(b) Some group of order 12 does not have a non-trivial proper normal subgroup.(c) Every group of order 12 has a subgroup of order 6.(d) Every group of order 12 has an element of order 12.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Which one of the following statements is true?(a) Every group of order 12 has a non-trivial proper normal subgroup.(b) Some group of order 12 does not have a non-trivial proper normal subgroup.(c) Every group of order 12 has a subgroup of order 6.(d) Every group of order 12 has an element of order 12.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which one of the following statements is true?(a) Every group of order 12 has a non-trivial proper normal subgroup.(b) Some group of order 12 does not have a non-trivial proper normal subgroup.(c) Every group of order 12 has a subgroup of order 6.(d) Every group of order 12 has an element of order 12.?.
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