AllTransitiveGroups - Maple Help

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GroupTheory

 TransitiveGroup
 compute the k-th transitive of a given degree
 NumTransitiveGroups
 compute the number of transitive groups of a given degree
 AllTransitiveGroups
 compute all the transitive groups of a given degree

 Calling Sequence TransitiveGroup( d, k ) NumTransitiveGroups( d ) AllTransitiveGroups( d )

Parameters

 d - : posint : the degree k - : posint : an index

Description

 • These three commands form an interface to the transitive groups database in the GroupTheory package.  Currently, conjugacy class representatives of the transitive groups up to degree $37$, but excluding those of degree $32$, are available in the database.
 • The NumTransitiveGroups( d ) command returns the number of transitive groups of degree d stored in the transitive groups database.  If the value returned is $0$, this indicates that no transitive groups of degree d are in the database.  (It does not indicate that there are no transitive groups of that degree!)
 • The TransitiveGroup( d, k ) command returns the $k$-th transitive group of degree $d$ from the transitive groups database.  If $k$ is larger than the number of transitive groups of degree d, an exception is raised.
 • The AllTransitiveGroups( d ) command returns a list of all the transitive groups of degree $d$.  It is essentially equivalent to [seq]( TransitiveGroup( d, k ), k = 1 .. NumTransitiveGroups( d ) ), but avoids some repeated checks.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{NumTransitiveGroups}\left(31\right)$
 ${12}$ (1)
 > $G≔\mathrm{TransitiveGroup}\left(31,3\right)$
 ${G}{≔}⟨\left({2}{,}{12}{,}{22}\right)\left({3}{,}{13}{,}{23}\right)\left({4}{,}{14}{,}{24}\right)\left({5}{,}{15}{,}{25}\right)\left({6}{,}{16}{,}{26}\right)\left({7}{,}{17}{,}{27}\right)\left({8}{,}{18}{,}{28}\right)\left({9}{,}{19}{,}{29}\right)\left({10}{,}{20}{,}{30}\right)\left({11}{,}{21}{,}{31}\right){,}\left({1}{,}{2}{,}{26}{,}{3}{,}{20}{,}{22}{,}{27}{,}{30}{,}{14}{,}{4}{,}{16}{,}{25}{,}{21}{,}{13}{,}{24}{,}{23}{,}{8}{,}{9}{,}{28}{,}{6}{,}{10}{,}{31}{,}{19}{,}{29}{,}{15}{,}{12}{,}{7}{,}{5}{,}{18}{,}{11}{,}{17}\right)⟩$ (2)
 > $\mathrm{IsTransitive}\left(G\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{IsPrimitive}\left(G\right)$
 ${\mathrm{true}}$ (4)
 > $L≔\mathrm{AllTransitiveGroups}\left(31\right):$$\mathrm{nops}\left(L\right)$
 ${12}$ (5)
 > $\mathrm{andmap}\left(\mathrm{IsTransitive},L\right)$
 ${\mathrm{true}}$ (6)

Transitive groups of prime degree are primitive.

 > $\mathrm{andmap}\left(\mathrm{IsPrimitive},L\right)$
 ${\mathrm{true}}$ (7)

For groups of non-prime degree, we normally find imprimitive groups.

 > $\mathrm{NumTransitiveGroups}\left(8\right)$
 ${50}$ (8)
 > $\mathrm{nops}\left(\mathrm{remove}\left(\mathrm{IsPrimitive},\mathrm{AllTransitiveGroups}\left(8\right)\right)\right)$
 ${43}$ (9)

Compatibility

 • The GroupTheory[TransitiveGroup], GroupTheory[NumTransitiveGroups] and GroupTheory[AllTransitiveGroups] commands were introduced in Maple 17.