Concept of Cyclic Group Video Lecture | Mathematics for Competitive Exams

A cyclic group is a mathematical structure consisting of a set of elements and an operation that satisfies certain properties. Specifically, a group is cyclic if there exists an element, called a generator, that can be repeatedly combined with itself (using the group operation) to produce all the other elements of the group. This means that every element in a cyclic group can be expressed as a power of the generator.

A cyclic group has several important properties. Firstly, it is closed under the group operation, meaning that combining any two elements in the group using the operation will always produce another element in the group. Secondly, it has an identity element, which is the element that, when combined with any other element, leaves that element unchanged. Thirdly, every element in a cyclic group has an inverse, meaning that there exists another element that, when combined with the original element, produces the identity element. Finally, a cyclic group is associative, meaning that the order in which elements are combined using the group operation does not affect the result.

A cyclic group can be represented in different ways. One common representation is using additive notation, where the group operation is denoted by '+'. In this representation, the generator is often denoted as 'g', and the elements of the group are expressed as powers of 'g'. For example, if 'g' is the generator of a cyclic group, the elements of the group could be written as g, g^2, g^3, and so on. Another representation is using multiplicative notation, where the group operation is denoted by '·' or '*'. In this representation, the generator is often denoted as 'a', and the elements of the group are expressed as powers of 'a'. For example, if 'a' is the generator of a cyclic group, the elements of the group could be written as a, a^2, a^3, and so on.

The order of a cyclic group is the number of elements it contains. In other words, it is the number of times the generator needs to be combined with itself using the group operation to produce all the elements of the group. The order of a cyclic group is always finite, and it can be any positive integer. For example, if a cyclic group has a generator 'g' and its order is 5, then the elements of the group would be g, g^2, g^3, g^4, and g^5.

No, a cyclic group can have only one generator. This is because the generator of a cyclic group is an element that, when combined with itself using the group operation, produces all the other elements of the group. If there were multiple generators, it would mean that there are multiple elements that can generate the entire group, which contradicts the definition of a cyclic group. Therefore, by definition, a cyclic group has only one generator.