### Quick description

This page defines several important group properties, providing examples for each one.

### Prerequisites

A basic knowledge of group theory, such as would be taught in typical first course on the subject.

#### Abelian groups

A group is abelian if the group operation is commutative; i.e., if for all . When a group is abelian, the operation is generally written additively as instead of multiplicatively as , and the identity is denoted as 0 instead of 1. A group that fails to be abelian is called nonabelian.

### Example 1

The integers with the usual addition operation form an abelian group. The same is true for any of the familiar number systems (, and ).

### Example 2

Since an element of a group commutes with all powers of itself, and a cyclic group consists of all the powers of a single element, it follows that cyclic groups are always abelian.

### Example 3

The group of invertible matrices with real entries forms a group under multiplication that is nonabelian.

#### Simple groups

A group is simple if it has no proper normal subgroups (that is, if its only normal subgroups are itself and the trivial group). Said another way, if is a surjective group homomorphism and is simple, then either is isomorphic to , or is the trivial group.

### Example 4

Since a cyclic group of prime order has no subgroups other than itself and the trivial group, it must be simple.

### Example 5

If is a finite abelian group, then any subgroup of is a normal subgroup of . Therefore, is simple if and only if it has no proper subgroups. This only happens when the order of is prime, in which case it is cyclic, as in the previous example.

#### Torsion groups

We say that a group is torsion if all of its elements have finite order. Of course, all finite groups are torsion, but there are many infinite torsion groups as well.

### Example 6

The additive group of is torsion: given any nonzero element , we can write it as , and so will be the integer . Thus will have order dividing in .

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