# Which group is torsion-free group?

Table of Contents

## Which group is torsion-free group?

abelian group

In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative and the identity element is the only element with finite order.

## What does it mean for a group to be torsion?

In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which each element has finite order. The exponent of such a group, if extant, is the least common multiple of the orders of the elements. The exponent exists for any finite group, and it divides the group order.

## Is free group Abelian?

The rank of a free abelian group is the cardinality of a basis; every two bases for the same group give the same rank, and every two free abelian groups with the same rank are isomorphic. The only free abelian groups that are free groups are the trivial group and the infinite cyclic group.

## Are torsion groups always finite?

Every finite abelian group is a torsion group. Not every torsion group is finite however: consider the direct sum of a countable number of copies of the cyclic group C2; this is a torsion group since every element has order 2.

## Are all torsion groups finite?

A torsion group (also called periodic group) is a group in which every element has finite order.

## How can you prove a group is free?

Definition 1 A group G is called a free group if there exists a generating set X of G such that every non-empty reduced group word in X defines a non-trivial element of G. In this event X is called a free basis of G and G is called free on X or freely generated by X.

## Are free groups Infinite?

A free group on a two-element set S occurs in the proof of the Banach–Tarski paradox and is described there. On the other hand, any nontrivial finite group cannot be free, since the elements of a free generating set of a free group have infinite order.