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Concept of Subgroup Video Lecture | Mathematics for Competitive Exams
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FAQs on Concept of Subgroup Video Lecture - Mathematics for Competitive Exams
| 1. What is a subgroup? |  |
Ans. A subgroup is a subset of a group that itself forms a group under the same operation as the original group. It contains the identity element of the group and is closed under the group operation.
| 2. How is a subgroup related to a group? | |
Ans. A subgroup is a subset of a group that retains the same group structure as the original group. It means that the subgroup follows the same group axioms as the group it is a part of, such as closure, associativity, identity element, and inverse element.
| 3. Can a subgroup have more elements than the original group? | |
Ans. No, a subgroup cannot have more elements than the original group. By definition, a subgroup is a subset of a group, which means it can only contain elements that are already present in the original group. However, a subgroup can have fewer elements or even the same number of elements as the original group.
| 4. How can we determine if a subset is a subgroup? | |
Ans. To determine if a subset is a subgroup, we need to check if it satisfies the subgroup criteria. The criteria are: (1) it contains the identity element of the group, (2) it is closed under the group operation, and (3) it contains the inverse element of each of its elements. If a subset meets these criteria, it is a subgroup.
| 5. Can a subgroup be empty? | |
Ans. No, a subgroup cannot be empty. By definition, a subgroup must contain the identity element of the group. Since the identity element is always present in a group, any subgroup must have at least one element, which is the identity element.
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Dec 23, 2024 Last updated |
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