Mathematics Exam > Mathematics Questions > Which of the following is TRUE ?a)If G is a f...
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Which of the following is TRUE ?
- a)If G is a finite group and a e G then 0 ( a ) / 0 ( G )
- b)The order of each subgroup of a finite group is a divisor of the order of the group
- c)A subgroup H of a cyclic group is also cyclic
- d)If aH = bH then aH and bH are disjoint
Correct answer is option 'A,B,C'. Can you explain this answer?
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Which of the following is TRUE ?a)If G is a finite group and a e G the...
Answer:
a) If G is a finite group and a ∈ G, then {0(a) mod 0(G)} is TRUE.
- This statement is true because for any finite group G and any element a in G, the order of the element a will always divide the order of the group G. Therefore, the result of taking the modulo of 0(a) by 0(G) will always be 0.
b) The order of each subgroup of a finite group is a divisor of the order of the group is TRUE.
- This statement is true because the order of a subgroup is defined as the number of elements in the subgroup. And since a subgroup is a subset of the original group, the number of elements in the subgroup must be a divisor of the number of elements in the original group.
c) A subgroup H of a cyclic group is also cyclic is TRUE.
- This statement is true because a cyclic group is generated by a single element, and any subgroup of a cyclic group will also be generated by a subset of that single element. Therefore, the subgroup itself will also be cyclic.
d) If aH = bH, then aH and bH are disjoint is FALSE.
- This statement is false because if aH = bH, it means that both a and b are in the same left coset of H. Therefore, their cosets aH and bH will have at least one element in common, which means they are not disjoint.
In summary, the true statements are:
a) If G is a finite group and a ∈ G, then {0(a) mod 0(G)} is true.
b) The order of each subgroup of a finite group is a divisor of the order of the group is true.
c) A subgroup H of a cyclic group is also cyclic is true.
The false statement is:
d) If aH = bH, then aH and bH are disjoint.
a) If G is a finite group and a ∈ G, then {0(a) mod 0(G)} is TRUE.
- This statement is true because for any finite group G and any element a in G, the order of the element a will always divide the order of the group G. Therefore, the result of taking the modulo of 0(a) by 0(G) will always be 0.
b) The order of each subgroup of a finite group is a divisor of the order of the group is TRUE.
- This statement is true because the order of a subgroup is defined as the number of elements in the subgroup. And since a subgroup is a subset of the original group, the number of elements in the subgroup must be a divisor of the number of elements in the original group.
c) A subgroup H of a cyclic group is also cyclic is TRUE.
- This statement is true because a cyclic group is generated by a single element, and any subgroup of a cyclic group will also be generated by a subset of that single element. Therefore, the subgroup itself will also be cyclic.
d) If aH = bH, then aH and bH are disjoint is FALSE.
- This statement is false because if aH = bH, it means that both a and b are in the same left coset of H. Therefore, their cosets aH and bH will have at least one element in common, which means they are not disjoint.
In summary, the true statements are:
a) If G is a finite group and a ∈ G, then {0(a) mod 0(G)} is true.
b) The order of each subgroup of a finite group is a divisor of the order of the group is true.
c) A subgroup H of a cyclic group is also cyclic is true.
The false statement is:
d) If aH = bH, then aH and bH are disjoint.
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Which of the following is TRUE ?a)If G is a finite group and a e G the...
(A), (B), (C) are correct statements. If two cosets aH and bH are equal then aH and bH are identical.
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Which of the following is TRUE ?a)If G is a finite group and a e G then 0 ( a ) / 0 ( G )b)The order of each subgroup of a finite group is a divisor of the order of the groupc)A subgroup H of a cyclic group is also cyclicd)If aH = bH then aH and bH are disjointCorrect answer is option 'A,B,C'. Can you explain this answer?
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Which of the following is TRUE ?a)If G is a finite group and a e G then 0 ( a ) / 0 ( G )b)The order of each subgroup of a finite group is a divisor of the order of the groupc)A subgroup H of a cyclic group is also cyclicd)If aH = bH then aH and bH are disjointCorrect answer is option 'A,B,C'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Which of the following is TRUE ?a)If G is a finite group and a e G then 0 ( a ) / 0 ( G )b)The order of each subgroup of a finite group is a divisor of the order of the groupc)A subgroup H of a cyclic group is also cyclicd)If aH = bH then aH and bH are disjointCorrect answer is option 'A,B,C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following is TRUE ?a)If G is a finite group and a e G then 0 ( a ) / 0 ( G )b)The order of each subgroup of a finite group is a divisor of the order of the groupc)A subgroup H of a cyclic group is also cyclicd)If aH = bH then aH and bH are disjointCorrect answer is option 'A,B,C'. Can you explain this answer?.
Which of the following is TRUE ?a)If G is a finite group and a e G then 0 ( a ) / 0 ( G )b)The order of each subgroup of a finite group is a divisor of the order of the groupc)A subgroup H of a cyclic group is also cyclicd)If aH = bH then aH and bH are disjointCorrect answer is option 'A,B,C'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Which of the following is TRUE ?a)If G is a finite group and a e G then 0 ( a ) / 0 ( G )b)The order of each subgroup of a finite group is a divisor of the order of the groupc)A subgroup H of a cyclic group is also cyclicd)If aH = bH then aH and bH are disjointCorrect answer is option 'A,B,C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following is TRUE ?a)If G is a finite group and a e G then 0 ( a ) / 0 ( G )b)The order of each subgroup of a finite group is a divisor of the order of the groupc)A subgroup H of a cyclic group is also cyclicd)If aH = bH then aH and bH are disjointCorrect answer is option 'A,B,C'. Can you explain this answer?.
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