Is there a finite group with an element of infinite order?

Is there a finite group with an element of infinite order?

Yes, in fact, the order of an element in a finite group divides the order of the group.

Can infinite groups have finite subgroups?

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.

How do you know if a group is finite?

If G is a finite group, every g ∈ G has finite order. The proof is as follows. Since the set of powers {ga : a ∈ Z} is a subset of G and the exponents a run over all integers, an infinite set, there must be a repetition: ga = gb for some a

Is every infinite group is Abelian?

An abelian group in which every element has finite oder is called a torsion abelian group; more generally, the subsets of elements of finite order form a subgroup called the torsion subgroup. There are also examples of infinite bounded abelian groups, such as (Z/nZ)N for some positive integer n.

Does there exist an infinite group having only finitely many subgroups justify your answer?

So G has infinitely many subgroups. If all elements of G has finite order then each are all finite and G = ∪g∈G is infinite. So there are infinitely many cyclic groups in G. Thus G has only finitely many subgroups.

Are finite groups cyclic?

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.

How many finite groups are there?

Summary. The following table is a complete list of the 18 families of finite simple groups and the 26 sporadic simple groups, along with their orders. Any non-simple members of each family are listed, as well as any members duplicated within a family or between families.

What is the difference between Infinity and infinite?

Infinite technically is an adjective. Like this page is white, similarly this series is infinite, but infinity is a noun. It means a certain quantity which is not a fixed number but bigger than any given number. Infinity ∞ is not a number, but an idea that describes something growing without bound.