Group: Examples

The sym­met­ric groups

For ev­ery pos­i­tive in­te­ger \(n\) there is a group \(S_n\), the sym­met­ric group of or­der \(n\), defined as the group of all per­mu­ta­tions (bi­jec­tions) \(\{ 1, 2, \dots n \} \to \{ 1, 2, \dots n \}\) (or any other set with \(n\) el­e­ments). The sym­met­ric groups play a cen­tral role in group the­ory: for ex­am­ple, a group ac­tion of a group \(G\) on a set \(X\) with \(n\) el­e­ments is the same as a ho­mo­mor­phism \(G \to S_n\).

Up to con­ju­gacy, a per­mu­ta­tion is de­ter­mined by its cy­cle type.

The dihe­dral groups

The dihe­dral groups \(D_{2n}\) are the col­lec­tions of sym­me­tries of an \(n\)-sided reg­u­lar poly­gon. It has a pre­sen­ta­tion \(\langle r, f \mid r^n, f^2, (rf)^2 \rangle\), where \(r\) rep­re­sents ro­ta­tion by \(\tau/n\) de­grees, and \(f\) rep­re­sents re­flec­tion.

For \(n > 2\), the dihe­dral groups are non-com­mu­ta­tive.

The gen­eral lin­ear groups

For ev­ery field \(K\) and pos­i­tive in­te­ger \(n\) there is a group \(GL_n(K)\), the gen­eral lin­ear group of or­der \(n\) over \(K\). Con­cretely, this is the group of all in­vert­ible \(n \times n\) ma­tri­ces with en­tries in \(K\); more ab­stractly, this is the au­to­mor­phism group of a vec­tor space of di­men­sion \(n\) over \(K\).

If \(K\) is alge­braically closed, then up to con­ju­gacy, a ma­trix is de­ter­mined by its Jor­dan nor­mal form.


  • Group

    The alge­braic struc­ture that cap­tures sym­me­try, re­la­tion­ships be­tween trans­for­ma­tions, and part of what mul­ti­pli­ca­tion and ad­di­tion have in com­mon.