Mathematics Exam > Mathematics Videos > Algebra > Concept of Subgroup
Concept of Subgroup Video Lecture | Algebra - Mathematics
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FAQs on Concept of Subgroup Video Lecture - Algebra - Mathematics
| 1. What is a subgroup? |  |
Ans. A subgroup is a subset of a group that maintains the same algebraic structure as the larger group. It is closed under the group operation, contains the identity element of the group, and contains the inverse of each of its elements.
| 2. How is a subgroup different from a group? | |
Ans. While a group is a complete set with a specific algebraic structure, a subgroup is a subset of a group that shares the same algebraic structure. A subgroup inherits its properties from the larger group, but it may have fewer elements and restrictions compared to the group itself.
| 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 contains fewer or an equal number of elements as the group itself.
| 4. How can we determine if a subset is a subgroup of a given group? | |
Ans. To determine if a subset is a subgroup of a given group, we need to check three conditions: closure, identity element, and inverses. The subset must be closed under the group operation, contain the identity element of the group, and contain the inverse of each of its elements.
| 5. Can a subgroup have multiple identity elements? | |
Ans. No, a subgroup cannot have multiple identity elements. The identity element is unique within a group, and a subgroup must contain the same identity element as the larger group. Therefore, a subgroup can only have one identity element.
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