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One-Step Subgroup Test Video Lecture | Mathematics for Competitive Exams
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FAQs on One-Step Subgroup Test Video Lecture - Mathematics for Competitive Exams
| 1. What is the One-Step Subgroup Test? |  |
Ans. The One-Step Subgroup Test is a method used in mathematics to determine if a subset of a group is a subgroup. It involves checking if the subset is closed under the group operation, contains the identity element, and contains the inverse of each of its elements.
| 2. How is the One-Step Subgroup Test applied in mathematics? | |
Ans. To apply the One-Step Subgroup Test, we need to check three conditions: closure, identity, and inverses. Firstly, we need to verify that the subset is closed under the group operation, meaning that if we combine any two elements from the subset, their result must also be in the subset. Secondly, the subset must contain the identity element of the group. Finally, for every element in the subset, its inverse must also be in the subset.
| 3. Can you provide an example of applying the One-Step Subgroup Test? | |
Ans. Sure! Let's consider the group of integers under addition, denoted as (Z, +). If we want to determine if the subset {0, 2, 4, 6} is a subgroup, we can apply the One-Step Subgroup Test. Firstly, we check closure: adding any two elements from the subset gives us another element in the subset. Secondly, the identity element of (Z, +) is 0, which is also in the subset. Lastly, for every element in the subset, its inverse is also in the subset. Therefore, {0, 2, 4, 6} satisfies all the conditions and is a subgroup of (Z, +).
| 4. What happens if a subset fails the One-Step Subgroup Test? | |
Ans. If a subset fails the One-Step Subgroup Test, it means that it is not a subgroup of the given group. This indicates that the subset lacks one or more of the required properties, such as closure, identity, or inverses. In such cases, the subset cannot be considered a subgroup.
| 5. Are there any alternative methods to determine if a subset is a subgroup? | |
Ans. Yes, there are alternative methods to determine if a subset is a subgroup. One common method is the Two-Step Subgroup Test, which involves verifying closure and inverses separately. Another approach is to use the Subgroup Criterion, which states that a non-empty subset of a group is a subgroup if and only if it is closed under the group operation and contains the inverse of each of its elements. Both of these methods can be used as alternatives to the One-Step Subgroup Test.
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Nov 22, 2024 Last updated |
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