# 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.

Parents:

• Group

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