# Is every abelian group finite?

## Is every abelian group finite?

To expand on my comment, all finite Abelian groups are finitely generated, but not all finitely generated Abelian groups are finite. An Abelian group ⟨G,+⟩ is finitely generated if there is a finite F⊆G such that every element of G can be written as the sum of elements of F or their inverses, possibly with repetition.

Is abelian group homomorphism?

A Group is Abelian if and only if Squaring is a Group Homomorphism Let G be a group and define a map f:G→G by f(a)=a2 for each a∈G. Then prove that G is an abelian group if and only if the map f is a group homomorphism. Proof. (⟹) If G is an abelian group, then f is a homomorphism.

### Are finite abelian groups finitely generated?

Every finite abelian group is finitely generated.

Is every abelian group is cyclic?

T F “Every abelian group is cyclic.” False: R and Q (under addition) and the Klein group V are all examples of abelian groups that are not cyclic.

## Is every Infinite abelian group is cyclic?

Not at all. If the abelian group is infinite, then, to be cyclic, it would have to be countable. And there are plenty uncountable abelian groups.

Does homomorphism preserve Abelian?

If φ : G → H is an isomorphism, prove that is abelian if and only if is abelian.

### Is finite Abelian group cyclic?

Every finite Abelian group is a direct product of cyclic groups of prime power order. Moreover, the number of terms in the product and the orders of the cyclic groups are uniquely determined by the group.

What do you mean by finite group?

In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations.

## Is finite abelian group cyclic?

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.

### What is finite group and infinite group?

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.

Is a group homomorphism always a group isomorphism?

We usually just say homomorphism and isomorphism instead of group homomorphism and group isomorphism. The identity map G→G G → G is an isomorphism. Let G,∗ and H,△ be any groups. The map f:G→H f : G → H given by f(g)=eH f ( g ) = e H for all g∈G g ∈ G is a group homomorphism.

## Is Z Z cyclic?

Now, in order for there to even be potential for an isomorphism, two spaces must have equal dimension. Since the dim(ZxZ)=2>dim(Z)=1, we know that ∄ an isomorphism between our spaces. Hence, ZxZ is not a cyclic group.