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Check the correct statement
- a)the identity of a subgroup is the same as that of the group.
- b)the inverse of any element of a subgroup is the same as the inverse o f the sam e regarded as an elem ent o f the group.
- c)the order of any element of a subgroup is the same as the order of the element regarded as a member of the group.
- d)All of the above.
Correct answer is option 'D'. Can you explain this answer?
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Check the correct statementa)the identity ofa subgroup is the same as ...
Explanation:
a) The identity of a subgroup is the same as that of the group.
The identity element in a group is an element that when combined with any other element in the group, leaves the other element unchanged. A subgroup is a subset of a group that itself forms a group under the same operation. Since a subgroup is a subset of the group, it contains the same identity element as the group. Therefore, the identity of a subgroup is the same as that of the group.
b) The inverse of any element of a subgroup is the same as the inverse of the same regarded as an element of the group.
In a group, every element has an inverse, which when combined with the element gives the identity element. A subgroup is a subset of a group, so the inverses of the elements in the subgroup are the same as the inverses of those elements in the group. Therefore, the inverse of any element of a subgroup is the same as the inverse of the same element regarded as an element of the group.
c) The order of any element of a subgroup is the same as the order of the element regarded as a member of the group.
The order of an element in a group is the smallest positive integer n such that raising the element to the power of n gives the identity element. Since a subgroup is a subset of a group, the order of an element in the subgroup is the same as the order of that element in the group. Therefore, the order of any element of a subgroup is the same as the order of the element regarded as a member of the group.
d) All of the above.
Since all of the statements a), b), and c) are true, the correct statement is d) All of the above.
a) The identity of a subgroup is the same as that of the group.
The identity element in a group is an element that when combined with any other element in the group, leaves the other element unchanged. A subgroup is a subset of a group that itself forms a group under the same operation. Since a subgroup is a subset of the group, it contains the same identity element as the group. Therefore, the identity of a subgroup is the same as that of the group.
b) The inverse of any element of a subgroup is the same as the inverse of the same regarded as an element of the group.
In a group, every element has an inverse, which when combined with the element gives the identity element. A subgroup is a subset of a group, so the inverses of the elements in the subgroup are the same as the inverses of those elements in the group. Therefore, the inverse of any element of a subgroup is the same as the inverse of the same element regarded as an element of the group.
c) The order of any element of a subgroup is the same as the order of the element regarded as a member of the group.
The order of an element in a group is the smallest positive integer n such that raising the element to the power of n gives the identity element. Since a subgroup is a subset of a group, the order of an element in the subgroup is the same as the order of that element in the group. Therefore, the order of any element of a subgroup is the same as the order of the element regarded as a member of the group.
d) All of the above.
Since all of the statements a), b), and c) are true, the correct statement is d) All of the above.
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Check the correct statementa)the identity ofa subgroup is the same as that of the group.b)the inverse of any element of a subgroup is the same as the inverse o f the sam e regarded as an elem ent o f the group.c)the order of any element of a subgroup is the same as the order of the elementregarded as a member of the group.d)All of the above.Correct answer is option 'D'. Can you explain this answer?
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Check the correct statementa)the identity ofa subgroup is the same as that of the group.b)the inverse of any element of a subgroup is the same as the inverse o f the sam e regarded as an elem ent o f the group.c)the order of any element of a subgroup is the same as the order of the elementregarded as a member of the group.d)All of the above.Correct answer is option 'D'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Check the correct statementa)the identity ofa subgroup is the same as that of the group.b)the inverse of any element of a subgroup is the same as the inverse o f the sam e regarded as an elem ent o f the group.c)the order of any element of a subgroup is the same as the order of the elementregarded as a member of the group.d)All of the above.Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Check the correct statementa)the identity ofa subgroup is the same as that of the group.b)the inverse of any element of a subgroup is the same as the inverse o f the sam e regarded as an elem ent o f the group.c)the order of any element of a subgroup is the same as the order of the elementregarded as a member of the group.d)All of the above.Correct answer is option 'D'. Can you explain this answer?.
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