Cyclic group | Cyclic group in group theory | Cyclic universe theory | Cyclic group mcqs

 

  A group G is said to be cyclic if every element of g is power of one and the same element a (say), of G.

CYCLIC GROUP:

DEFINITION:

                     A group G is said to be cyclic if every element of g is power of one and the same element a (say), of G.

                or

                    A cyclic group is a group that can be generated by a single element. (the group generator). Cyclic groups are Abelian
            such an element a of G is called a generator of G.
            if the order of a a is finite, that is, if there is least positive integer n such that a^n=e, then G is said to be a finite cyclic group of order n and written as 
                        

                                G=  < a : a^n = e >

(read as g is a cyclic group of order n generated by a).

        if the order of a is finite then, for no positive integer n, a^n = e

EXAMPLE:

                    For example, (Z/6Z)^× = {1,5}, and since 6 is twice an odd prime this is a cyclic group.. When (Z/nZ)^× is cyclic, its generators are called primitive roots modulo n. For a prime number p, the group (Z/pZ)^× is always cyclic, consisting of the non-zero elements of the finite field of order p.

How do you know if a group is cyclic?

A finite group is cyclic if, and only if, it has precisely one subgroup of each divisor of its order. So if you find two subgroups of the same order, then the group is not cyclic, and that can help sometimes.

Cosets of subgroup H in group G:

Let $H$ be a subgroup of a group $G$. If $a \in G$, the right coset of $H$ generated by $a$ is, $Ha$ = { $ha, h ∈ H$ };
and similarly, the left coset of $H$ generated by $a$ is $aH$ = { $ah, h ∈ H$ }

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