Do cyclic groups have finite order?

Related classes of groups Every cyclic group is virtually cyclic, as is every finite group. An infinite group is virtually cyclic if and only if it is finitely generated and has exactly two ends; an example of such a group is the direct product of Z/nZ and Z, in which the factor Z has finite index n.

What is a finite cyclic group?

Definition. A finite cyclic group is a group satisfying the following equivalent conditions: It is both finite and cyclic. It is isomorphic to the group of integers modulo n for some positive integer .

How many generators does a cyclic group of order n have?

of generators of the cyclic group of order n will be euler’s tautient function ϕ(n)…. that is no of m such that mfour generators of the group…

How do you prove a cyclic is a finite group?

Let G be a group. If the order of G is a prime, or of the form pq where p numbers such that pdoes not divide , then G is cyclic.

When we say an element in a group has infinite order?

In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is infinite.

Can a finite group have an element of infinite order?

If the group is of finite order then the order of every element in the group divides the order of the group. Hence no element can have infinite order. In your example, if pq+Z∈Q/Z then (pq+Z)q=q(pq+Z)=Z which is the identity element of this group. Hence every element is of finite order.

What is a cyclic group in cryptography?

A cyclic group G is a group that can be generated by a single element a , so that every element in G has the form ai for some integer i . We denote the cyclic group of order n by Zn , since the additive group of Zn is a cyclic group of order n .

What is the total number of generators of a finite cyclic group of order 28?

Therefore, there are 12 generators of G.