Sage-Tips

Is finite Abelian group?

A finite abelian group is a group satisfying the following equivalent conditions: It is isomorphic to a direct product of finitely many finite cyclic groups. It is isomorphic to a direct product of abelian groups of prime power order. It is isomorphic to a direct product of cyclic groups of prime power order.

What is finite and infinite abelian group?

This chapter describes the methods and results of infinite abelian groups. A group is Abelian if the group operation, usually called “addition,” is commutative. The theory of Abelian groups is an independent branch of algebra. Each finite Abelian group is a direct sum of cyclic groups of prime power orders.

What is finite non Abelian group?

In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (G, ∗) in which there exists at least one pair of elements a and b of G, such that a ∗ b ≠ b ∗ a. It is the smallest finite non-abelian group.

Can an abelian group be infinite?

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. Thus what you are looking for are infinite torsion abelian groups.

Is finite abelian group cyclic?

Every finite abelian group is an internal group direct product of cyclic groups whose orders are prime powers. The number of terms in the product and the orders of the cyclic groups are uniquely determined by the group.

Are all infinite 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.

Are all finite groups cyclic?

What is the rank of a finite abelian group?

The finite abelian group is just the torsion subgroup of G. The rank of G is defined as the rank of the torsion-free part of G; this is just the number n in the above formulas. A corollary to the fundamental theorem is that every finitely generated torsion-free abelian group is free abelian.

What is the fundamental theorem of abelian groups?

Stated differently the fundamental theorem says that a finitely generated abelian group is the direct sum of a free abelian group of finite rank and a finite abelian group, each of those being unique up to isomorphism. The finite abelian group is just the torsion subgroup of G.

Is every finite abelian group isomorphic to a direct product?

Every finite abelian group is isomorphic to a direct product of cyclic groups of prime power order; that is, every finite abelian group is isomorphic to a group of the type where each p k is prime (not necessarily distinct).

What is the difference between abelian and commutative rings?

Every ring is an abelian group with respect to its addition operation. In a commutative ring the invertible elements, or units, form an abelian multiplicative group. In particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication.