Can an infinite group has a finite subgroup?

Then the direct sum of the groups in S is an infinite group in which every element has finite order, so generates a finite subgroup. (The direct sum is the set of sequences for which the ith entry is a member of the ith group in S, such that all but a finite number of the entries is the identity of its group.

Can an infinite group have 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.

What are finite and infinite groups?

As we know that if a set has a starting point and an ending point both, it is a finite set, but it is infinite if it has no end from any side or both sides. If a set has the unlimited number of elements, then it is infinite and if the elements are countable then it is finite.

What is an example of an infinite group?

Let us recall a few examples of infinite groups we have seen: the group of real numbers (with addition), • the group of complex numbers (with addition), • the group of rational numbers (with addition).

Can a cyclic group be infinite?

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.

Is Q Z cyclic?

The group ℚ/ℤ is an injective object in the category Ab of abelian groups. It is also a cogenerator in the category of abelian groups. But if ⟨a⟩ is the cyclic subgroup generated by a, then it is easy to find a map g:⟨a⟩→ℚ/ℤ such that g(a)≠0, and then we can extend g to a map f:A→ℚ/ℤ using injectivity of ℚ/ℤ.

How do you know if its finite or infinite?

Points to determine a set as finite or infinite are:

1. If a set has a starting and end point both then it is finite but if it does not have a starting or end point then it is infinite set.
2. If a set has a limited number of elements then it is finite but if its number of elements is unlimited then it is infinite.

Is 0 a finite number?

Zero is a finite number. When we say that a number is infinite, it means that it is uncountable, limitless, or endless.

Is Z an infinite group?

However, in Z all elements are of infinite order, except for 0. But (as you have shown) in Q/Z there are many elements of various finite orders. Since order of elements is preserved under an isomorphism it is impossible for Q/Z to be isomorphic to Z, and so the former is not cyclic.

What is order of infinite group?

If no such m exists, a is said to have infinite order. The order of a group G is denoted by ord(G) or |G|, and the order of an element a is denoted by ord(a) or |a|. The order of an element a is equal to the order of its cyclic subgroup ⟨a⟩ = {ak for k an integer}, the subgroup generated by a. Thus, |a| = |⟨a⟩|.

Is Z * Z cyclic?

So Z × Z cannot be cyclic. Alternative method: draw a picture of Z×Z and 〈(n, m)〉 for a typical element (n, m) ∈ Z×Z and show that 〈(n, m)〉 is contained in the straight line mx = ny, so can’t cover all of Z × Z (since there’s no single straight line containing all of the points in the plane with integer coordinates).

Is Q Z cyclic group?

https://www.youtube.com/watch?v=YAK5HZ2N-s4