The SearchTransitiveGroups( spec ) command searches Maple's transitive groups database for groups satisfying properties specified in a sequence spec of search parameters. This allows you to locate examples of transitive permutation groups that have specific, supported properties, or combinations of those properties.

Use the form = X option to control the form of the output from this command. By default, an expression sequence of IDs for the TransitiveGroups database is returned. This is the same as specifying form = "id". To have an expression sequence of permutation groups, use the form = "permgroup"  option. The form = "count" option causes SearchTransitiveGroups to return just the number of groups in the database satisfying the constraints implied by the search parameters.

Note that the IDs returned in the default case are the IDs of the groups within the TransitiveGroups database.  These may differ from the IDs for the same group if it happens to be present in another database, which has its own set of group IDs. In particular, the first member of the TransitiveGroups database ID is the degree of the group, not its order.

So, for example, the symmetric group of degree $3$ appears in the database of transitive groups with ID equal to (3, 2), but appears also in the database of small groups with ID equal to (6, 1).

The valid search parameters may be grouped into several classes, as follows.

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

$\mathrm{SearchTransitiveGroups}\left('\mathrm{form}'=''count''\right)$

$162212$

$\mathrm{id}≔\mathrm{SearchTransitiveGroups}\left(\mathrm{degree}=6\,\mathrm{transitivity}=2\right)$

$\mathrm{id}≔\left[6\,12\right]$

$G≔\mathrm{TransitiveGroup}\left(\mathrm{op}\left(\mathrm{id}\right)\right)$

$G≔⟨\left(1\,2\,3\,4\,6\right)\,\left(1\,4\right)\left(5\,6\right)⟩$

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

$6$

$\mathrm{id}≔\mathrm{SearchTransitiveGroups}\left(\mathrm{order}=24\&comma;1<\mathrm{transitivity}\right)$

$\mathrm{id}≔\left[4\&comma;5\right]$

$\mathrm{Transitivity}\left(\mathrm{TransitiveGroup}\left(\mathrm{op}\left(\mathrm{id}\right)\right)\right)$

$4$

$\mathrm{SearchTransitiveGroups}\left(\mathrm{degree}=6\&comma;\mathrm{isregular}\&comma;\mathrm{form}=\mathrm{permgroup}\right)$

$⟨\left(1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\right)⟩,⟨\left(1\&comma;3\&comma;5\right)\left(2\&comma;4\&comma;6\right)\&comma;\left(1\&comma;4\right)\left(2\&comma;3\right)\left(5\&comma;6\right)⟩$

$\mathrm{SearchTransitiveGroups}\left(\mathrm{isprimitive}\&comma;\mathrm{form}=\mathrm{count}\right)$

$288$

$\mathrm{SearchTransitiveGroups}\left(\mathrm{degree}=24\&comma;\mathrm{form}=\mathrm{count}\right)=\mathrm{NumTransitiveGroups}\left(24\right)$

$25000=25000$

The GroupTheory[SearchTransitiveGroups] command was introduced in Maple 2015.

For more information on Maple 2015 changes, see Updates in Maple 2015.

The GroupTheory[SearchTransitiveGroups] command was updated in Maple 2019.