ElementaryGroup - Maple Help

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GroupTheory

 ElementaryGroup

 Calling Sequence ElementaryGroup( p, n ) ElementaryGroup( p, n, form = f )

Parameters

 p - algebraic; understood to represent a prime number n - algebraic; understood to represent a positive integer f - string ; either "permgroup" or "fpgroup"

Description

 • An elementary Abelian group of order ${p}^{n}$, where $p$ is a prime number, is a direct product (or direct sum) of $n$ copies of the cyclic group of order $p$.
 • The ElementaryGroup( p, n ) command returns an elementary Abelian group of order ${p}^{n}$, either as a permutation group or as a finitely presented group, according to the value of the form option. By default, a permutation group is returned.
 • If either of p and n is not an explicit integer, then a symbolic group representing the elementary group of order ${p}^{n}$ is returned.
 • In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{ElementaryGroup}\left(3,4\right)$
 ${G}{≔}{{C}}_{{3}}^{{4}}$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${81}$ (2)
 > $\mathrm{IsAbelian}\left(G\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{Generators}\left(G\right)$
 $\left[\left({1}{,}{2}{,}{3}\right){,}\left({4}{,}{5}{,}{6}\right){,}\left({7}{,}{8}{,}{9}\right){,}\left({10}{,}{11}{,}{12}\right)\right]$ (4)
 > $G≔\mathrm{ElementaryGroup}\left(3,5,'\mathrm{form}'="fpgroup"\right)$
 ${G}{≔}{{C}}_{{3}}^{{5}}$ (5)
 > $G≔\mathrm{ElementaryGroup}\left(17,3\right)$
 ${G}{≔}{{C}}_{{17}}^{{3}}$ (6)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${4913}$ (7)
 > $\mathrm{GroupOrder}\left(\mathrm{ElementaryGroup}\left(p,n+m\right)\right)$
 ${{p}}^{{n}{+}{m}}$ (8)
 > $\mathrm{IsCyclic}\left(\mathrm{ElementaryGroup}\left(p,n\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}1
 ${\mathrm{false}}$ (9)

Compatibility

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