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Group Theory: Cyclic Group Video Lecture | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
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FAQs on Group Theory: Cyclic Group Video Lecture - Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
| 1. What is a cyclic group? |  |
Ans. A cyclic group is a mathematical concept in group theory that represents a group with a single generator. It is a group that can be generated by a single element, called a generator, by repeatedly applying the group operation.
| 2. How do you determine if a group is cyclic? | |
Ans. To determine if a group is cyclic, we need to check if there exists an element in the group that can generate the entire group by repeated application of the group operation. If such an element exists, the group is cyclic; otherwise, it is not.
| 3. Can a cyclic group have more than one generator? | |
Ans. No, a cyclic group can have only one generator. The generator is an element that can generate the entire group by repeated application of the group operation. If there were multiple generators, it would imply that different elements can generate the group, which contradicts the definition of a cyclic group.
| 4. Are all subgroups of a cyclic group also cyclic? | |
Ans. Yes, all subgroups of a cyclic group are also cyclic. This property is known as the cyclic subgroup property. In other words, if a group is cyclic and contains a subgroup, that subgroup is also cyclic. The generator of the subgroup can be obtained by taking a power of the generator of the original cyclic group.
| 5. Can a non-cyclic group have a cyclic subgroup? | |
Ans. Yes, a non-cyclic group can have a cyclic subgroup. While the group as a whole may not be generated by a single element, it is still possible for a subgroup within the group to be cyclic. This means that there exists an element in the subgroup that can generate the subgroup by repeated application of the group operation.
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