DirectFactors - Maple Help

GroupTheory

 DirectFactors
 compute the directly indecomposable direct factors of a finite group
 IsDirectlyIndecomposable
 determine if a finite group is directly indecomposable

 Calling Sequence DirectFactors( G ) IsDirectlyIndecomposable( G )

Parameters

 G - a finite group

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Alt}\left(4\right)$
 ${G}{≔}{{\mathbf{A}}}_{{4}}$ (1)
 > $\mathrm{DirectFactors}\left(G\right)$
 ${{\mathbf{A}}}_{{4}}$ (2)
 > $\mathrm{IsDirectlyIndecomposable}\left(\mathrm{Alt}\left(4\right)\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{df}≔\mathrm{DirectFactors}\left(\mathrm{GL}\left(2,4\right)\right)$
 ${\mathrm{df}}{≔}⟨\left({1}{,}{2}{,}{3}\right)\left({4}{,}{8}{,}{12}\right)\left({5}{,}{10}{,}{15}\right)\left({6}{,}{11}{,}{13}\right)\left({7}{,}{9}{,}{14}\right)⟩{,}⟨\left({1}{,}{8}{,}{13}{,}{14}{,}{10}\right)\left({2}{,}{12}{,}{6}{,}{7}{,}{15}\right)\left({3}{,}{4}{,}{11}{,}{9}{,}{5}\right){,}\left({1}{,}{12}{,}{5}{,}{6}{,}{14}\right)\left({2}{,}{4}{,}{10}{,}{11}{,}{7}\right)\left({3}{,}{8}{,}{15}{,}{13}{,}{9}\right)⟩$ (4)
 > $\mathrm{AreIsomorphic}\left(\mathrm{GL}\left(2,4\right),\mathrm{DirectProduct}\left(\mathrm{df}\right)\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{IsDirectlyIndecomposable}\left(\mathrm{df}\left[1\right]\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{IsDirectlyIndecomposable}\left(\mathrm{df}\left[2\right]\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{IsDirectlyIndecomposable}\left(\mathrm{GL}\left(2,4\right)\right)$
 ${\mathrm{false}}$ (8)
 > $\mathrm{df}≔\left[\mathrm{DirectFactors}\right]\left(\mathrm{CyclicGroup}\left(24\right)\right)$
 ${\mathrm{df}}{≔}\left[⟨\left({1}{,}{4}{,}{7}{,}{10}{,}{13}{,}{16}{,}{19}{,}{22}\right)\left({2}{,}{5}{,}{8}{,}{11}{,}{14}{,}{17}{,}{20}{,}{23}\right)\left({3}{,}{6}{,}{9}{,}{12}{,}{15}{,}{18}{,}{21}{,}{24}\right)⟩{,}⟨\left({1}{,}{9}{,}{17}\right)\left({2}{,}{10}{,}{18}\right)\left({3}{,}{11}{,}{19}\right)\left({4}{,}{12}{,}{20}\right)\left({5}{,}{13}{,}{21}\right)\left({6}{,}{14}{,}{22}\right)\left({7}{,}{15}{,}{23}\right)\left({8}{,}{16}{,}{24}\right)⟩\right]$ (9)
 > $\mathrm{AreIsomorphic}\left(\mathrm{CyclicGroup}\left(24\right),\mathrm{DirectProduct}\left(\mathrm{op}\left(\mathrm{df}\right)\right)\right)$
 ${\mathrm{true}}$ (10)
 > $\mathrm{IsDirectlyIndecomposable}\left(\mathrm{CyclicGroup}\left(27\right)\right)$
 ${\mathrm{true}}$ (11)
 > $\mathrm{seq}\left(\mathrm{IsDirectlyIndecomposable}\left(\mathrm{DihedralGroup}\left(n\right)\right),n=2..10\right)$
 ${\mathrm{false}}{,}{\mathrm{true}}{,}{\mathrm{true}}{,}{\mathrm{true}}{,}{\mathrm{false}}{,}{\mathrm{true}}{,}{\mathrm{true}}{,}{\mathrm{true}}{,}{\mathrm{false}}$ (12)
 > $\mathrm{IsDirectlyIndecomposable}\left(\mathrm{WreathProduct}\left(\mathrm{Symm}\left(3\right),\mathrm{CyclicGroup}\left(2\right)\right)\right)$
 ${\mathrm{true}}$ (13)
 > $\mathrm{IsDirectlyIndecomposable}\left(\mathrm{FrobeniusGroup}\left(100,1\right)\right)$
 ${\mathrm{true}}$ (14)

Compatibility

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