Properties of groups

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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 article 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.[/note]

==== 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]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]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]The group of invertible $2 \times 2$ matrices with real entries forms a group under multiplication that is nonabelian.[/EXAMPLE]

==== 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 $\varphi: G \to H$ is a surjective group homomorphism and $G$ is simple, then either $H$ is isomorphic to $G$, or $H$ is the trivial group.

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

[EXAMPLE] 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.[/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] 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 $\Q/\Z$.[/EXAMPLE]

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