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Elementary Properties of Groups Video Lecture | Mathematics for Competitive Exams
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FAQs on Elementary Properties of Groups Video Lecture - Mathematics for Competitive Exams
| 1. What are the four elementary properties of groups? |  |
Ans. The four elementary properties of groups are closure, associativity, identity element, and inverse element. Closure means that the operation between any two elements of the group will always result in another element within the group. Associativity means that the order in which the operations are performed does not affect the final result. Identity element is an element in the group that, when combined with any other element, leaves the other element unchanged. Inverse element is an element in the group that, when combined with another element, results in the identity element.
| 2. How does closure property ensure that a group is closed under the operation? | |
Ans. The closure property ensures that a group is closed under the operation by guaranteeing that the result of the operation between any two elements of the group will always be another element within the group. This means that for any two elements, a and b, in the group, their operation, denoted as a * b, will also be an element of the group. If the group is not closed under the operation, it cannot be considered a group.
| 3. What is the significance of the identity element in a group? | |
Ans. The identity element in a group is significant because it serves as a neutral element for the group's operation. When any element of the group is combined with the identity element, it remains unchanged. In other words, the identity element acts as a "do-nothing" element. It ensures that every element in the group has an element it can combine with to produce itself. The identity element is unique within the group and is an essential property of a group.
| 4. How does the inverse element relate to the identity element in a group? | |
Ans. The inverse element in a group is closely related to the identity element. For every element a in a group, there exists an element b, called the inverse of a, such that the operation between a and b results in the identity element of the group. In other words, a * b = b * a = identity element. The inverse element "undoes" the operation with a particular element and brings it back to the neutral state represented by the identity element.
| 5. Can you provide an example of a group that satisfies all the elementary properties? | |
Ans. Yes, one example of a group that satisfies all the elementary properties is the set of integers under addition, denoted as (Z, +). The closure property holds because the sum of any two integers is always an integer. Associativity holds because the order of addition does not affect the result. The identity element is 0, as adding 0 to any integer leaves the integer unchanged. Finally, every integer has an inverse element, where the inverse of an integer x is -x, as x + (-x) = 0.
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Dec 23, 2024 Last updated |
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