Can an infinite group have a finite subgroup?

Can an infinite group have a finite subgroup?

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

How do you prove two infinite groups are isomorphic?

Proof: By definition, two groups are isomorphic if there exist a 1-1 onto mapping ϕ from one group to the other. In order for us to have 1-1 onto mapping we need that the number of elements in one group equal to the number of the elements of the other group. Thus, the two groups must have the same order.

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.

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.

Does there exist an infinite group with only a finite number of subgroups?

No. An infinite group either contains Z, which has infinitely many subgroups, or each element has finite order, but then the union G=⋃g∈G⟨g⟩ must be made of infinitely many subgroups. Note that there are infinite groups with only a finite number of normal subgroups.

Is there an infinite group all of whose proper subgroups are finite?

No. The direct limit of the cyclic groups of order pn is infinite, but every proper subgroup of it is finite.

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.

Which of the following is example of finite group?

A finite group is a group having finite group order. Examples of finite groups are the modulo multiplication groups, point groups, cyclic groups, dihedral groups, symmetric groups, alternating groups, and so on.

Can infinite groups be isomorphic?

All infinite cyclic groups are isomorphic. That is, up to isomorphism, there is only one infinite cyclic group.

How many infinite groups are there?

How many infinite groups are there? Infinitely many. And certainly as many as you can ever produce as a large cardinal number.