# The Smallest Abelian group Ever Found

Monday, October 6th 2014. | Science

If you like to study algebra or mathematics, you have to know the smallest Abelian group.  Many people also call the Abelian group as a commutative group in abstract algebra. The addition of integers in the arithmetic is generalized in the Abelian groups. The concept of Abelian groups was derived from the name of Niels Henrik Abel. If you compare the theory and concept of the Abelian groups, you can find them simpler that the non Abelian group.

## The Smallest Abelian Group: Klein four-group

Klein four-group is called as the smallest Abelian group. Some people often call it as Vierergruppe or Klein group. You can write the group with V symbol. This group is defined as a group of Z2 x Z2. The name of the group was derived from Felix Klein. He called the group as Vierergruppe in his book with the title Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade. It was published in 1884. The bit strings of Klein four-group include {00, 01, 10, and 11}.

Klein four-group book

### The Smallest Abelian Group: the simplest infinite Abelian group

Let’s check out the simplest infinite Abelian group. The infinite cyclic group Z is called as the simplest one. The torsion free groups and torsion groups are called as the opposite properties. However, both are included as two important classes in the smallest Abelian group.

Klein four-group pic

## The Smallest Abelian Group: Relation to other mathematical topics

The Abelian group has many relations with other mathematical topics. You can find out the natural topology as the nature of many largest Abelian groups. Therefore, you can turn the Abelian groups into the topological groups.

Klein four-group

### The Smallest Abelian Group: the prototype of Abelian category

Have you ever checked the prototype of the Abelian category? It is called as the category of Ab.  The Abelian group included in the category is the ones which can be formed with homomorphisms.

smallest abelian group

## The Smallest Abelian Group: Tarski’s student Szmielew

Tarski’s student Szmielew in 1955 made a surprising breakthrough by defining the first order theory of the Abelian groups. Many of the members in the Abelian group are decidable. Therefore, the Abelian groups are easier to define than the non Abelian groups. The people can conduct further theories and researches for the Abelian groups due to the decidability. There are many successes in the usage and application of the Abelian group since they are easier to decide.

smallest abelian groups

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