For an element $g$ of finite order in a group $G$, the prime power factors of $g$ are elements $g_1,g_2,..,g_k$ of $G$ such that $g=g_1·g_2..g_k$, and such that each $g_i$ has order equal to a power of a prime number. The elements $g_i$ are pairwise commutative, and are uniquely determined up to the order in which they occur.

The PrimePowerFactors( g, G ) command computes the prime power factors of the group element $g$.

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

$f≔\mathrm{PrimePowerFactors}\left(\mathrm{Perm}\left(\left[\left[1\,2\,3\,4\,5\,6\right]\right]\right)\,\mathrm{Symm}\left(6\right)\right)$

$f≔\left(1\,4\right)\left(2\,5\right)\left(3\,6\right),\left(1\,5\,3\right)\left(2\,6\,4\right)$

$\mathrm{andmap}\left(\mathrm{type}\,\mathrm{map}\left(\mathrm{ElementOrder}\,\left[f\right]\,\mathrm{Symm}\left(6\right)\right)\,'\mathrm{primepower}'\right)$

$\mathrm{true}$

$\mathrm{PermProduct}\left(f\right)$

$\left(1\,2\,3\,4\,5\,6\right)$

The GroupTheory[PrimePowerFactors] command was introduced in Maple 2019.

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