IsElementary - Maple Help

GroupTheory

 IsElementary
 attempt to determine whether a group is elementary Abelian

 Calling Sequence IsElementary( G )

Parameters

 G - a finite group

Description

 • A group $G$ is elementary if it is a finite Abelian group with prime exponent. Equivalently, $G$ is elementary if it is a direct sum (product) of groups each of order equal to a fixed prime $p$.
 • The IsElementary( G ) command attempts to determine whether the group G is elementary.  It returns true if G is elementary and returns false otherwise.
 • The group G must be an instance of a permutation group, a Cayley table group or a finite, finitely presented group.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{SmallGroup}\left(32,1\right):$
 > $\mathrm{IsElementary}\left(G\right)$
 ${\mathrm{false}}$ (1)
 > $G≔\mathrm{SmallGroup}\left(17,1\right):$
 > $\mathrm{IsElementary}\left(G\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{IsElementary}\left(\mathrm{DirectProduct}\left(\mathrm{}\left(\mathrm{CyclicGroup}\left(2\right),5\right)\right)\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{IsElementary}\left(\mathrm{WreathProduct}\left(\mathrm{}\left(\mathrm{CyclicGroup}\left(2\right),5\right)\right)\right)$
 ${\mathrm{false}}$ (4)
 > $G≔\mathrm{CayleyTableGroup}\left(⟨⟨⟨1|2|3|4⟩,⟨2|1|4|3⟩,⟨3|4|1|2⟩,⟨4|3|2|1⟩⟩⟩\right)$
 ${G}{≔}{\mathrm{< a Cayley table group with 4 elements >}}$ (5)
 > $\mathrm{IsElementary}\left(G\right)$
 ${\mathrm{true}}$ (6)
 > $G≔\mathrm{CayleyTableGroup}\left(⟨⟨⟨1|2|3|4|5|6⟩,⟨2|1|4|3|6|5⟩,⟨3|5|1|6|2|4⟩,⟨4|6|2|5|1|3⟩,⟨5|3|6|1|4|2⟩,⟨6|4|5|2|3|1⟩⟩⟩\right)$
 ${G}{≔}{\mathrm{< a Cayley table group with 6 elements >}}$ (7)
 > $\mathrm{IsElementary}\left(G\right)$
 ${\mathrm{false}}$ (8)
 > $\mathrm{IsElementary}\left(⟨⟨a,b,c⟩|⟨{a}^{5},{b}^{5},{c}^{5},a·b=b·a,a·c=c·a,b·c=c·b⟩⟩\right)$
 ${\mathrm{true}}$ (9)

Compatibility

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