Can a finite group G have an element G with infinite order?

When G is a finite group, every element must have finite order. However, the converse is false: there are infinite groups where each element has finite order. Let g ∈ G and g have order n. Then gk = e if and only if n | k.

Is every element in a finite group is of finite order?

The order of every element of a finite group is finite and is less than or equal to the order of the group. Proof : Suppose G is a finite group, the composition being denoted multiplicatively.

When a group is of finite 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. 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 group of finite order 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.

Can a finite group have an infinite subgroup?

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

Does an infinite group have an 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.

What is the order of G?

The Order of a group (G) is the number of elements present in that group, i.e it’s cardinality. It is denoted by |G|. Order of element a ∈ G is the smallest positive integer n, such that an= e, where e denotes the identity element of the group, and an denotes the product of n copies of a.

What is G in group theory?

In abstract algebra, the center of a group, G, is the set of elements that commute with every element of G. It is denoted Z(G), from German Zentrum, meaning center.

Is every finite group a cyclic group?

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 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.

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.

How do you prove G is not a finite group?

Since S is finite, there exist positive integers n < m such that a n = a m . Hence a m − n = 1. Let a be an element of G . If a is of infinite order, G contains a subgroup of infinite order. Hence G is not a finite group.

What is the Order of every element of a finite group?

Theorem 1: The order of every element of a finite group is finite. a, a 2, a 3, a 4, … Every one of these powers must be an element of G.

What is the Order of an element of a group theorem?

Theorems on the Order of an Element of a Group Theorem 1:The order of every element of $$a$$ finite group is finite. Proof:Let $$G$$ be a finite group and let $$a \\in G$$, we consider all positive integral powers of $$a$$, i.e. \\[a,{a^2},{a^3},{a^4},…\\]

Is the Order of an element always less than the group?

Yes, the order of an element is always less than or equal to the order of the group. In the proof above, assume n 1 and n 2 are all positive and that n 1 < n 2. Find the least such pair n 1 and n 2. If n 1 is greater than the order of the group, then that meant you saw at least n 1 different thing before seeing a repeat.