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GroupTheory

 AlternatingGroup

 Calling Sequence AlternatingGroup( n, formopt ) Alt( n, formopt )

Parameters

 n - algebraic; understood to be a positive integer formopt - (optional) equation of the form form = F, where F is either "permgroup" (the default) or "fpgroup"

Description

 • The alternating group ${\mathbf{A}}_{n}$ on $n$ elements is the set of all even permutations of$\left\{1,2,\dots ,n\right\}$ for a positive integer $n$. The order of ${\mathbf{A}}_{n}$ is equal to $\frac{n!}{2}$, for $1. The alternating group of degree $n$ is simple if $n$ is at least $5$.
 • The AlternatingGroup( n ) command returns an alternating permutation group of degree n.  You can also use Alt( n ) as an abbreviation of AlternatingGroup( n ).
 • The form = F option controls the form of the group returned. By default, a permutation group is returned; this is equivalent to passing the option form = "permgroup". A finitely presented group can be obtained by passing the option form = "fpgroup".
 • If the argument n is not an integer constant, then a symbolic group is returned. In this case, the form option is ignored.
 • In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{AlternatingGroup}\left(7\right)$
 ${G}{≔}{{\mathbf{A}}}_{{7}}$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${2520}$ (2)
 > $\mathrm{IsTransitive}\left(G\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{IsPrimitive}\left(G\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{IsSimple}\left(G\right)$
 ${\mathrm{true}}$ (5)
 > $G≔\mathrm{Alt}\left(4\right)$
 ${G}{≔}{{\mathbf{A}}}_{{4}}$ (6)
 > $\mathrm{IsSimple}\left(G\right)$
 ${\mathrm{false}}$ (7)
 > $\mathrm{DrawSubgroupLattice}\left(G\right)$ > $G≔\mathrm{Alt}\left(5,'\mathrm{form}'="fpgroup"\right)$
 ${G}{≔}⟨{}{s}{,}{t}{}{\mid }{}{{s}}^{{3}}{,}{{t}}^{{3}}{,}{t}{}{{s}}^{{-1}}{}{t}{}{s}{}{t}{}{{s}}^{{-1}}{}{t}{}{s}{,}{s}{}{t}{}{s}{}{t}{}{s}{}{t}{}{s}{}{t}{}{s}{}{t}{}⟩$ (8)

If the argument to the constructor is not a literal integer, then a symbolic group is returned.

 > $G≔\mathrm{Alt}\left(3n+7\right)$
 ${G}{≔}{{\mathbf{A}}}_{{3}{}{n}{+}{7}}$ (9)
 > $\mathrm{IsSimple}\left(G\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}n::'\mathrm{posint}'$
 ${\mathrm{true}}$ (10)
 > $\mathrm{GroupOrder}\left(G\right)$
 $\frac{\left({3}{}{n}{+}{7}\right){!}}{{2}}$ (11)

Compatibility

 • The GroupTheory[AlternatingGroup] command was introduced in Maple 17.