Can an infinite group have an element of finite order?

There are infinitely many rational numbers in [0,1), and hence the order of the group Q/Z is infinite. Thus the order of the element mn+Z is at most n. Hence the order of each element of Q/Z is finite. Therefore, Q/Z is an infinite group whose elements have finite orders.

Can an infinite group has a finite subgroup?

In conclusion: Zp∞ is an infinite group whose proper subgroups are all finite. It is a non-cyclic group whose all proper subgroups are cyclic.

What is finite and infinite group?

1. Finite versus Infinite Groups and Elements: Groups may be broadly categorized in a number of ways. One is simply how large the group is. (a) Definition: The order of a group G, denoted |G|, is the number of elements in a group. This is either a finite number or is infinite.

Can a group have infinite order?

If a group is not finite, one says that its order is infinite. The order of an element of a group (also called period length or period) is the order of the subgroup generated by the element.

What is an example of an infinite group?

The most basic example of an infinite group is the set of integers . It is a group under addition, and its identity is 0. It is the (up to isomorphism) unique infinite cyclic group.

Is every infinite group is Abelian?

Consider any polynomial rings over any finite field. If we strip away its multiplication, we get an infinite abelian group with respect to the addition. Every non-zero element has order p, the characteristic of the underlying field.

Can a group have two elements?

A group can not contain exactly two elements of order 2.

When can you say that a group has an infinite number of elements?

If any set is endless from start or end or both sides having continuity then we can say that set is infinite. For example, the set of whole numbers, W = {0, 1, 2, 3, ……..} is an infinite set as the number of elements is infinite.