Properties of groups

This means that it cannot be considered to contain or lead to any mathematically interesting information.

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.

This article is incomplete. Properties to include: solvable, perfect. What other properties are worth including? I plan to organize a table at the end showing, for each set of properties, a (nontrivial) group or family of groups satisfying that exact set of properties.

Abelian groups

A group $G$ is abelian if the group operation is commutative; i.e., if $gh = hg$ for all $g,h \in G$ . When a group is abelian, the operation is generally written additively as $g + h$ instead of multiplicatively as $gh$ , and the identity is denoted as 0 instead of 1. A group that fails to be abelian is called nonabelian.

Example 1

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

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 $2 \times 2$ matrices with real entries forms a group under multiplication that is nonabelian.

Simple groups

A group $G$ 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 $G \to H$ is a surjective group homomorphism and $G$ is simple, then either $H$ is isomorphic to $G$ , or $H$ 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 $G$ is a finite abelian group, then any subgroup $H$ of $G$ is a normal subgroup of $G$ . Therefore, $G$ is simple if and only if it has no proper subgroups. This only happens when the order of $G$ is prime, in which case it is cyclic, as in the previous example.

Torsion groups

We say that a group $G$ 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 $\Q/\Z$ is torsion: given any nonzero element $g \in \Q$ , we can write it as $p/q$ , and so $qg$ will be the integer $p$ . Thus $g$ will have order dividing $q$ in $\Q/\Z$ .