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Which of the following is/are true for G to be an abelian group?
(a) a^2= e, 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?
(a) a^2= e, 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?(a) a^...
Understanding Abelian Groups
An abelian group is defined as a group in which the group operation is commutative, meaning that for any two elements a and b in the group G, the equation ab = ba holds. Let's analyze the given statements:
(a) a^2 = e, forall a ∈ G
- This implies that every element is its own inverse. While this ensures that the group is abelian for elements of order 2, it does not guarantee commutativity for all groups. Therefore, this condition is not sufficient for G to be an abelian group.
(b) o(a) = 2, forall a in G
- Here, "o(a)" represents the order of element a, meaning each element has order 2. This means a^2 = e for all a in G, which is similar to statement (a). Again, while it implies a certain structure, it doesn't guarantee that the group is abelian. Thus, this condition is not sufficient.
(c) ab = ba, forall a, b ∈ G
- This is the definitive property of abelian groups. If this condition is satisfied, G is indeed an abelian group. Hence, this statement is true.
(d) (ab)^-1 = (b^-1 * a^-1), forall a, b in G
- This is a general property of groups and holds for all groups, not just abelian ones. It states that the inverse of a product is the product of the inverses in reverse order. Therefore, while this is true, it does not imply that G is abelian.
Conclusion
- Only statement (c) correctly defines an abelian group. Statements (a) and (b) provide conditions that may occur in abelian groups but do not guarantee commutativity. Statement (d) is universally true for all groups.
An abelian group is defined as a group in which the group operation is commutative, meaning that for any two elements a and b in the group G, the equation ab = ba holds. Let's analyze the given statements:
(a) a^2 = e, forall a ∈ G
- This implies that every element is its own inverse. While this ensures that the group is abelian for elements of order 2, it does not guarantee commutativity for all groups. Therefore, this condition is not sufficient for G to be an abelian group.
(b) o(a) = 2, forall a in G
- Here, "o(a)" represents the order of element a, meaning each element has order 2. This means a^2 = e for all a in G, which is similar to statement (a). Again, while it implies a certain structure, it doesn't guarantee that the group is abelian. Thus, this condition is not sufficient.
(c) ab = ba, forall a, b ∈ G
- This is the definitive property of abelian groups. If this condition is satisfied, G is indeed an abelian group. Hence, this statement is true.
(d) (ab)^-1 = (b^-1 * a^-1), forall a, b in G
- This is a general property of groups and holds for all groups, not just abelian ones. It states that the inverse of a product is the product of the inverses in reverse order. Therefore, while this is true, it does not imply that G is abelian.
Conclusion
- Only statement (c) correctly defines an abelian group. Statements (a) and (b) provide conditions that may occur in abelian groups but do not guarantee commutativity. Statement (d) is universally true for all groups.
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Which of the following is/are true for G to be an abelian group?(a) a^2= e, 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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