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Cyclic Group Video Lecture | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
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FAQs on Cyclic Group Video Lecture - Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
| 1. What is a cyclic group? |  |
A cyclic group is a mathematical structure consisting of a set of elements, along with an operation that combines any two elements to produce a third element. The group is said to be cyclic if there exists an element, called the generator, such that repeatedly applying the operation to the generator produces all the elements in the group.
| 2. How can we determine if a group is cyclic? | |
To determine if a group is cyclic, we need to check if there exists an element in the group that generates all other elements. This can be done by repeatedly applying the operation to a chosen element and checking if all elements are obtained. If such an element exists, the group is cyclic; otherwise, it is not.
| 3. What is the order of a cyclic group? | |
The order of a cyclic group is the number of elements it contains. In other words, it is the total count of distinct elements that can be generated by repeatedly applying the operation to the generator. The order is always a positive integer.
| 4. Can a cyclic group have more than one generator? | |
No, a cyclic group can have only one generator. This is because the generator must be able to produce all other elements in the group through repeated application of the operation. If there were multiple generators, one generator would not be able to generate all the elements produced by the other generators.
| 5. Are all cyclic groups finite? | |
No, not all cyclic groups are finite. While some cyclic groups have a finite number of elements, there are also cyclic groups that have an infinite number of elements. The order of a cyclic group can be finite (e.g., a cyclic group of integers modulo n) or infinite (e.g., a cyclic group of integers).
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Dec 22, 2024 Last updated |
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