The DirectFactors( G ) command computes an expression sequence of subgroups of the finite permutation group G such that G is the (internal) direct product of these subgroups, and each subgroup is directly indecomposable.

The Remak-Krull-Schmidt Theorem guarantees that the decomposition is unique up to isomorphism and ordering of the direct factors.

A group is indecomposable if it has no proper non-trivial direct factor. The IsDirectlyIndecomposable( G ) command returns true if G is indecomposable, and false otherwise.

The group G must be an instance of a permutation group.

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

$G≔\mathrm{Alt}\left(4\right)$

$G≔{\mathbf{A}}_4$

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

${\mathbf{A}}_4$

$\mathrm{IsDirectlyIndecomposable}\left(\mathrm{Alt}\left(4\right)\right)$

$\mathrm{true}$

$\mathrm{df}≔\mathrm{DirectFactors}\left(\mathrm{GL}\left(2\,4\right)\right)$

$\mathrm{df}≔⟨\left(1\,2\,3\right)\left(4\,8\,12\right)\left(5\,10\,15\right)\left(6\,11\,13\right)\left(7\,9\,14\right)⟩,⟨\left(1\,8\,13\,14\,10\right)\left(2\,12\,6\,7\,15\right)\left(3\,4\,11\,9\,5\right)\,\left(1\,12\,5\,6\,14\right)\left(2\,4\,10\,11\,7\right)\left(3\,8\,15\,13\,9\right)⟩$

$\mathrm{AreIsomorphic}\left(\mathrm{GL}\left(2\,4\right)\,\mathrm{DirectProduct}\left(\mathrm{df}\right)\right)$

$\mathrm{true}$

$\mathrm{IsDirectlyIndecomposable}\left(\mathrm{df}\left[1\right]\right)$

$\mathrm{true}$

$\mathrm{IsDirectlyIndecomposable}\left(\mathrm{df}\left[2\right]\right)$

$\mathrm{true}$

$\mathrm{IsDirectlyIndecomposable}\left(\mathrm{GL}\left(2\,4\right)\right)$

$\mathrm{false}$

$\mathrm{df}≔\left[\mathrm{DirectFactors}\right]\left(\mathrm{CyclicGroup}\left(24\right)\right)$

$\mathrm{df}≔\left[⟨\left(1\,4\,7\,10\,13\,16\,19\,22\right)\left(2\,5\,8\,11\,14\,17\,20\,23\right)\left(3\,6\,9\,12\,15\,18\,21\,24\right)⟩\,⟨\left(1\,9\,17\right)\left(2\,10\,18\right)\left(3\,11\,19\right)\left(4\,12\,20\right)\left(5\,13\,21\right)\left(6\,14\,22\right)\left(7\,15\,23\right)\left(8\,16\,24\right)⟩\right]$

$\mathrm{AreIsomorphic}\left(\mathrm{CyclicGroup}\left(24\right)\,\mathrm{DirectProduct}\left(\mathrm{op}\left(\mathrm{df}\right)\right)\right)$

$\mathrm{true}$

$\mathrm{IsDirectlyIndecomposable}\left(\mathrm{CyclicGroup}\left(27\right)\right)$

$\mathrm{true}$

$\mathrm{seq}\left(\mathrm{IsDirectlyIndecomposable}\left(\mathrm{DihedralGroup}\left(n\right)\right)\,n=2..10\right)$

$\mathrm{false},\mathrm{true},\mathrm{true},\mathrm{true},\mathrm{false},\mathrm{true},\mathrm{true},\mathrm{true},\mathrm{false}$

$\mathrm{IsDirectlyIndecomposable}\left(\mathrm{WreathProduct}\left(\mathrm{Symm}\left(3\right)\,\mathrm{CyclicGroup}\left(2\right)\right)\right)$

$\mathrm{true}$

$\mathrm{IsDirectlyIndecomposable}\left(\mathrm{FrobeniusGroup}\left(100\,1\right)\right)$

$\mathrm{true}$

The GroupTheory[DirectFactors] and GroupTheory[IsDirectlyIndecomposable] commands were introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.