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.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{SmallGroup}\left(32\,1\right)\:$

$\mathrm{IsElementary}\left(G\right)$

$\mathrm{false}$

$G≔\mathrm{SmallGroup}\left(17\,1\right)\:$

$\mathrm{IsElementary}\left(G\right)$

$\mathrm{true}$

$\mathrm{IsElementary}\left(\mathrm{DirectProduct}\left(\mathrm{`\$`}\left(\mathrm{CyclicGroup}\left(2\right)\,5\right)\right)\right)$

$\mathrm{true}$

$\mathrm{IsElementary}\left(\mathrm{WreathProduct}\left(\mathrm{`\$`}\left(\mathrm{CyclicGroup}\left(2\right)\,5\right)\right)\right)$

$\mathrm{false}$

$G≔\mathrm{CayleyTableGroup}\left(⟨⟨⟨1\left|2\right|3|4⟩\,⟨2\left|1\right|4|3⟩\,⟨3\left|4\right|1|2⟩\,⟨4\left|3\right|2|1⟩⟩⟩\right)$

$G≔\mathrm{<\; a\; Cayley\; table\; group\; with\; 4\; elements\; >}$

$\mathrm{IsElementary}\left(G\right)$

$\mathrm{true}$

$G≔\mathrm{CayleyTableGroup}\left(⟨⟨⟨1\left|2\right|3\left|4\right|5|6⟩\&comma;⟨2\left|1\right|4\left|3\right|6|5⟩\&comma;⟨3\left|5\right|1\left|6\right|2|4⟩\&comma;⟨4\left|6\right|2\left|5\right|1|3⟩\&comma;⟨5\left|3\right|6\left|1\right|4|2⟩\&comma;⟨6\left|4\right|5\left|2\right|3|1⟩⟩⟩\right)$

$G≔\mathrm{<\; a\; Cayley\; table\; group\; with\; 6\; elements\; >}$

$\mathrm{IsElementary}\left(G\right)$

$\mathrm{false}$

$\mathrm{IsElementary}\left(⟨⟨a\&comma;b\&comma;c⟩|⟨a^5\&comma;b^5\&comma;c^5\&comma;a·b=b·a\&comma;a·c=c·a\&comma;b·c=c·b⟩⟩\right)$

$\mathrm{true}$

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

For more information on Maple 17 changes, see Updates in Maple 17.