Can a finite group have an infinite subgroup?

Can a finite group have an infinite subgroup?

A subgroup of a finite group is finite. Done. let G be a finite group of order n. It cannot have an element of infinite order since a belongs to G implies a^n=e and so o(a) is any positive integer less than or equal to n.

Can a subgroup have infinite order?

The order of an element of a group (also called period length or period) is the order of the subgroup generated by the element. If no such m exists, the order of a is infinite.

How many subgroups does an infinite cyclic group have?

The infinite cyclic group There is one subgroup dZ for each integer d (consisting of the multiples of d), and with the exception of the trivial group (generated by d = 0) every such subgroup is itself an infinite cyclic group.

Can you have a finite subgroup of an infinite group?

An infinite group G may have subgroups H of finite index (for example, the even integers inside the group of integers). Such a subgroup always contains a normal subgroup N (of G), also of finite index.

Is a subgroup of an infinite group infinite?

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.

Can finite groups have elements with infinite order?

Yes, A group can have an element of finite order and infinite order simultaneously.. For example the group of integers (Z,+). Thus, if G1 is your favorite non-trivial finite group and G2 is your favorite infinite group, then G=G1×G2 has both elements of finite order and elements of infinite order.

Are all 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 subgroups does an infinite group have?

If G is an infinite group then G has infinitely many subgroups. Proof: Let’s consider the following set: C={⟨g⟩:g∈G} – collection of all cyclic subgroups in G generated by elements of G. Two cases are possible: Exists infinitely many distinct cyclic subgroups ⇒ We are done.

Can a cyclic group be infinite?

How do you identify subgroups?

The most basic way to figure out subgroups is to take a subset of the elements, and then find all products of powers of those elements. So, say you have two elements a,b in your group, then you need to consider all strings of a,b, yielding 1,a,b,a2,ab,ba,b2,a3,aba,ba2,a2b,ab2,bab,b3,…

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