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Which of the following is/are true for G to be an abelian group?
(X) is not
(a) ae, forall a \in G
(b) o(a) = 2 , forall a in G
(c) ab = ba , forall a be G
(d) ( ab)^ - 1 = (b ^ -1 * a^-1) , forall a,b in G?
(X) is not
(a) ae, forall a \in G
(b) o(a) = 2 , forall a in G
(c) ab = ba , forall a be G
(d) ( ab)^ - 1 = (b ^ -1 * a^-1) , forall a,b in G?
Most Upvoted Answer
Which of the following is/are true for G to be an abelian group?(X) is...
Understanding Abelian Groups
In group theory, an abelian group is defined by the commutative property of its operation. Let's analyze each statement regarding the conditions for G to be an abelian group.
Statement Analysis
- (a) ae, forall a ∈ G
- This statement appears incomplete or unclear. Typically, it might suggest that "ae" is an identity element, but this does not directly relate to commutativity. Thus, this statement does not guarantee G is abelian.
- (b) o(a) = 2, forall a ∈ G
- Here, o(a) refers to the order of element a. If every element has order 2, then a * a = e (identity element). However, while this implies a certain structure, it does not necessarily establish commutativity. Hence, this statement alone does not ensure G is abelian.
- (c) ab = ba, forall a, b ∈ G
- This is the definition of an abelian group. If for all elements a and b in G, the equation ab = ba holds, then G is indeed an abelian group.
- (d) (ab) ^ -1 = (b ^ -1 * a ^ -1), forall a, b ∈ G
- This statement is true for all groups due to the properties of group inverses. It does not depend on the group being abelian. Therefore, while it is a valid property of groups, it does not confirm that G is abelian.
Conclusion
To summarize, among the given statements, only statement (c) directly confirms that G is an abelian group. Statements (a), (b), and (d) do not provide sufficient conditions to establish G as abelian.
In group theory, an abelian group is defined by the commutative property of its operation. Let's analyze each statement regarding the conditions for G to be an abelian group.
Statement Analysis
- (a) ae, forall a ∈ G
- This statement appears incomplete or unclear. Typically, it might suggest that "ae" is an identity element, but this does not directly relate to commutativity. Thus, this statement does not guarantee G is abelian.
- (b) o(a) = 2, forall a ∈ G
- Here, o(a) refers to the order of element a. If every element has order 2, then a * a = e (identity element). However, while this implies a certain structure, it does not necessarily establish commutativity. Hence, this statement alone does not ensure G is abelian.
- (c) ab = ba, forall a, b ∈ G
- This is the definition of an abelian group. If for all elements a and b in G, the equation ab = ba holds, then G is indeed an abelian group.
- (d) (ab) ^ -1 = (b ^ -1 * a ^ -1), forall a, b ∈ G
- This statement is true for all groups due to the properties of group inverses. It does not depend on the group being abelian. Therefore, while it is a valid property of groups, it does not confirm that G is abelian.
Conclusion
To summarize, among the given statements, only statement (c) directly confirms that G is an abelian group. Statements (a), (b), and (d) do not provide sufficient conditions to establish G as abelian.
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Which of the following is/are true for G to be an abelian group?(X) is not(a) ae, forall a \in G(b) o(a) = 2 , forall a in G(c) ab = ba , forall a be G(d) ( ab)^ - 1 = (b ^ -1 * a^-1) , forall a,b in G?
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Which of the following is/are true for G to be an abelian group?(X) is not(a) ae, forall a \in G(b) o(a) = 2 , forall a in G(c) ab = ba , forall a be G(d) ( ab)^ - 1 = (b ^ -1 * a^-1) , forall a,b in G? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Which of the following is/are true for G to be an abelian group?(X) is not(a) ae, forall a \in G(b) o(a) = 2 , forall a in G(c) ab = ba , forall a be G(d) ( ab)^ - 1 = (b ^ -1 * a^-1) , forall a,b in G? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following is/are true for G to be an abelian group?(X) is not(a) ae, forall a \in G(b) o(a) = 2 , forall a in G(c) ab = ba , forall a be G(d) ( ab)^ - 1 = (b ^ -1 * a^-1) , forall a,b in G?.
Which of the following is/are true for G to be an abelian group?(X) is not(a) ae, forall a \in G(b) o(a) = 2 , forall a in G(c) ab = ba , forall a be G(d) ( ab)^ - 1 = (b ^ -1 * a^-1) , forall a,b in G? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Which of the following is/are true for G to be an abelian group?(X) is not(a) ae, forall a \in G(b) o(a) = 2 , forall a in G(c) ab = ba , forall a be G(d) ( ab)^ - 1 = (b ^ -1 * a^-1) , forall a,b in G? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following is/are true for G to be an abelian group?(X) is not(a) ae, forall a \in G(b) o(a) = 2 , forall a in G(c) ab = ba , forall a be G(d) ( ab)^ - 1 = (b ^ -1 * a^-1) , forall a,b in G?.
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