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Group Theory: Subgroup Video Lecture | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
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FAQs on Group Theory: Subgroup Video Lecture - Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
| 1. What is a subgroup in group theory? |  |
A subgroup is a subset of a group that forms a group itself. It satisfies all the axioms of a group, namely closure, associativity, identity element, and inverse element. In other words, a subgroup is a smaller group that is contained within a larger group.
| 2. How can we determine if a subset is a subgroup? | |
To determine if a subset is a subgroup, we need to check if it satisfies the necessary conditions. These conditions are: (1) closure under the group operation, (2) associativity, (3) existence of an identity element, and (4) existence of inverse elements for every element in the subset. If all these conditions are met, then the subset is a subgroup.
| 3. Can a subgroup have a different operation than the original group? | |
No, a subgroup always uses the same operation as the original group. The group operation is inherited by the subgroup, meaning that the operation performed on elements of the subgroup is the same as the operation performed on elements of the original group. This ensures that the subgroup is a valid group itself.
| 4. What is the significance of subgroups in group theory? | |
Subgroups play a crucial role in the study of group theory. They allow us to break down a larger group into smaller, more manageable parts, which can often reveal important properties and symmetries. Subgroups also help us classify and analyze different types of groups, as many group properties can be understood by examining the subgroups they contain.
| 5. Can a group have multiple subgroups? | |
Yes, a group can have multiple subgroups. In fact, every group has at least two subgroups: the trivial subgroup, which consists of just the identity element, and the group itself. Additionally, there can be non-trivial subgroups that are proper subsets of the group, which means they are not equal to the group itself. The number and nature of subgroups a group has depend on its specific properties and structure.
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Dec 18, 2024 Last updated |
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