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GroupTheory

 PrimePowerFactors
 factor a group element as a product of elements of prime power order

 Calling Sequence PrimePowerFactors( g, G )

Parameters

 g - element of G whose factorization is to be computed G - group containing the element g

Description

 • 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 $g$.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $f≔\mathrm{PrimePowerFactors}\left(\mathrm{Perm}\left(\left[\left[1,2,3,4,5,6\right]\right]\right),\mathrm{Symm}\left(6\right)\right)$
 ${f}{≔}\left({1}{,}{4}\right)\left({2}{,}{5}\right)\left({3}{,}{6}\right){,}\left({1}{,}{5}{,}{3}\right)\left({2}{,}{6}{,}{4}\right)$ (1)
 > $\mathrm{andmap}\left(\mathrm{type},\mathrm{map}\left(\mathrm{ElementOrder},\left[f\right],\mathrm{Symm}\left(6\right)\right),'\mathrm{primepower}'\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{PermProduct}\left(f\right)$
 $\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}\right)$ (3)

Compatibility

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