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Properties of groups

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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.

Note iconIncomplete 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.