opt : equation of the form 'output' = value, where value is either list (the default) or iterator

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( r ) command returns a list of all the transitive groups of degree $r$ if $r$ is a positive integer.  It is essentially equivalent to [seq]( TransitiveGroup( d, k ), k = 1 .. NumTransitiveGroups( d ) ), but avoids some repeated checks. If $r$ is a range of the form a .. b, then AllTransitiveGroups( r ) returns the transitive groups of degree $d$, for a <= d <= b. By default, or if the output = list option is passed, then a list of the requested groups is returned. If you pass the output = iterator option, then an iterator object is returned.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\&colon;$

$\mathrm{NumTransitiveGroups}\left(31\right)$

$12$

$G≔\mathrm{TransitiveGroup}\left(31\&comma;3\right)$

$G≔⟨\left(2\&comma;12\&comma;22\right)\left(3\&comma;13\&comma;23\right)\left(4\&comma;14\&comma;24\right)\left(5\&comma;15\&comma;25\right)\left(6\&comma;16\&comma;26\right)\left(7\&comma;17\&comma;27\right)\left(8\&comma;18\&comma;28\right)\left(9\&comma;19\&comma;29\right)\left(10\&comma;20\&comma;30\right)\left(11\&comma;21\&comma;31\right)\&comma;\left(1\&comma;13\&comma;7\&comma;14\&comma;31\&comma;3\&comma;8\&comma;11\&comma;25\&comma;15\&comma;27\&comma;6\&comma;2\&comma;24\&comma;5\&comma;4\&comma;19\&comma;20\&comma;9\&comma;17\&comma;21\&comma;12\&comma;30\&comma;10\&comma;26\&comma;23\&comma;18\&comma;16\&comma;29\&comma;22\&comma;28\right)⟩$

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

$\mathrm{true}$

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

$\mathrm{true}$

$L≔\mathrm{AllTransitiveGroups}\left(31\right)\&colon;$$\mathrm{nops}\left(L\right)$

$12$

$\mathrm{andmap}\left(\mathrm{IsTransitive}\&comma;L\right)$

$\mathrm{true}$

$\mathrm{andmap}\left(\mathrm{IsPrimitive}\&comma;L\right)$

$\mathrm{true}$

$\mathrm{NumTransitiveGroups}\left(8\right)$

$50$

$\mathrm{nops}\left(\mathrm{remove}\left(\mathrm{IsPrimitive}\&comma;\mathrm{AllTransitiveGroups}\left(8\right)\right)\right)$

$43$

$\mathrm{AllTransitiveGroups}\left(2..4\right)$

$\left[⟨\left(1\&comma;2\right)⟩\&comma;⟨\left(1\&comma;2\&comma;3\right)⟩\&comma;⟨\left(2\&comma;3\right)\&comma;\left(1\&comma;2\&comma;3\right)⟩\&comma;⟨\left(1\&comma;2\&comma;3\&comma;4\right)⟩\&comma;⟨\left(1\&comma;2\right)\left(3\&comma;4\right)\&comma;\left(1\&comma;4\right)\left(2\&comma;3\right)⟩\&comma;⟨\left(2\&comma;4\right)\&comma;\left(1\&comma;2\&comma;3\&comma;4\right)⟩\&comma;⟨\left(2\&comma;4\&comma;3\right)\&comma;\left(1\&comma;3\right)\left(2\&comma;4\right)⟩\&comma;⟨\left(2\&comma;4\&comma;3\right)\&comma;\left(1\&comma;4\&comma;2\&comma;3\right)⟩\right]$

$\mathrm{it}≔\mathrm{AllTransitiveGroups}\left(24\&comma;'\mathrm{output}'=''iterator''\right)$

$\mathrm{it}≔\mathrm{\&lang;Transitive\; Groups\; Iterator:\; 24/1\; ..\; 24/25000\&rang;}$

$i≔0\&colon;$

$$\begin{gathered} \mathbf{for}G\mathbf{in}\mathrm{it}\mathbf{do} \\ \mathrm{`+=`}\left(i\right); \\ \mathbf{if}\mathbf{not}\mathrm{IsRegular}\left(G\right)\mathbf{then} \\ \mathbf{break} \\ \mathbf{end}\mathbf{if} \\ \mathbf{end}\mathbf{do}\&colon; \end{gathered}$$

$i$

$16$

$\mathrm{it}≔\mathrm{AllTransitiveGroups}\left(31..33\&comma;'\mathrm{output}'=''iterator''\right)$

$\mathrm{it}≔\mathrm{\&lang;Transitive\; Groups\; Iterator:\; 31/1\; ..\; 33/162\&rang;}$

$$\begin{gathered} \mathbf{for}\mathrm{id},G\mathbf{in}\mathrm{it}\mathbf{do} \\ \mathbf{if}\mathbf{not}\mathrm{IsPrimitive}\left(G\right)\mathbf{then} \\ \mathrm{print}\left(\mathrm{id}\right); \\ \mathbf{break} \\ \mathbf{end}\mathbf{if} \\ \mathbf{end}\mathbf{do}\&colon; \end{gathered}$$

$33,1$

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

For more information on Maple 17 changes, see Updates in Maple 17.

The GroupTheory[TransitiveGroup], GroupTheory[NumTransitiveGroups] and GroupTheory[AllTransitiveGroups] commands were updated in Maple 2022.

The r parameter was updated in Maple 2022.

The output option was introduced in Maple 2022.

For more information on Maple 2022 changes, see Updates in Maple 2022.