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.

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

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

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

$\mathrm{false}$

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

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

$\mathrm{true}$

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

$\mathrm{true}$

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

$\mathrm{false}$

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

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

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

$\mathrm{true}$

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

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

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

$\mathrm{false}$

$\mathrm{IsElementary}\left(⟨⟨a\&comma;b\&comma;c⟩\&verbar;⟨a^5\&comma;b^5\&comma;c^5\&comma;\mathrm{`.`}\left(a\&comma;b\right)\&equals;\mathrm{`.`}\left(b\&comma;a\right)\&comma;\mathrm{`.`}\left(a\&comma;c\right)\&equals;\mathrm{`.`}\left(c\&comma;a\right)\&comma;\mathrm{`.`}\left(b\&comma;c\right)\&equals;\mathrm{`.`}\left(c\&comma;b\right)⟩⟩\right)$

$\mathrm{true}$

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

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