SearchTransitiveGroups - Maple Help

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GroupTheory

 SearchTransitiveGroups
 search for transitive groups satisfying specified properties

 Calling Sequence SearchTransitiveGroups( spec, formopt )

Parameters

 spec - expression sequence of search parameters formopt - (optional) an option of the form form = X, where X is one of "id" (the default), "permgroup", or "count".

Description

 • 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.

Boolean Search Parameters

 • Boolean search parameters p, such as isregular, can be specified in one of the forms p = true, p = false or just p, which is equivalent to p = true. If the boolean search parameter p is true, then only groups satisfying the corresponding predicate are returned.  If the boolean search parameter p is false, then only groups that do not satisfy the predicate are returned. Leaving a boolean search parameter unspecified causes the SearchTransitiveGroups command to return groups that do, and do not, satisfy the corresponding predicate.
 • Two boolean search parameters are currently supported, and are described in the following table.

 primitive describes the class of primitive groups regular describes the class of regular groups frobenius describes the class of Frobenius groups abelian describes the class of Abelian groups nilpotent describes the class of nilpotent groups soluble describes the class of soluble groups perfect describes the class of perfect groups simple describes the class of simple groups almostsimple describes the class of almostsimple groups

 • A transitive permutation group is primitive if it does not admit a non-trivial system of blocks.  Equivalently, a transitive group is primitive if some (hence, all) point stabilizers are maximal subgroups.
 • A permutation group is regular if it is transitive and no non-trivial element has a fixed point.

Numeric Search Parameters

 • Maple supports three search parameters that describe numeric invariants of transitive groups. Each has a positive integer value. A numeric search parameter p may be given in the form p = n, for some specific value n, or by indicating a range, as in p = a .. b. In the former case, only groups for which the numeric parameter has the value n will be returned. In the case in which a range is specified, groups for which the numeric invariant lies within the indicate range (inclusive of its end-points) are returned.
 • The supported numeric search parameters are listed in the following table.

 degree indicates the degree of the group order indicates the order (cardinality) of the group transitivity indicates the transitivity of the group

 • A permutation group $G$ is said to be $k$-transitive if it is transitive and if the stabilizer of a point is $k-1$-transitive on the remaining points.  A permutation group is $1$-transitive if it is transitive. Groups that are $2$-transitive are said to be doubly transitive and a triply transitive group is one that is $3$-transitive. The transitivity of $G$ is the largest value of $k$ for which $G$ is $k$-transitive.

Examples

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

Count the total number of groups in the database.

 > $\mathrm{SearchTransitiveGroups}\left('\mathrm{form}'="count"\right)$
 ${162212}$ (1)

Find the doubly-transitive groups of degree 6

 > $\mathrm{id}≔\mathrm{SearchTransitiveGroups}\left(\mathrm{degree}=6,\mathrm{transitivity}=2\right)$
 ${\mathrm{id}}{≔}\left[{6}{,}{12}\right]$ (2)
 > $G≔\mathrm{TransitiveGroup}\left(\mathrm{op}\left(\mathrm{id}\right)\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{3}{,}{4}{,}{6}\right){,}\left({1}{,}{4}\right)\left({5}{,}{6}\right)⟩$ (3)
 > $\mathrm{Degree}\left(G\right)$
 ${6}$ (4)

Which transitive groups in the database have order 24 and are multiply transitive?

 > $\mathrm{id}≔\mathrm{SearchTransitiveGroups}\left(\mathrm{order}=24,1<\mathrm{transitivity}\right)$
 ${\mathrm{id}}{≔}\left[{4}{,}{5}\right]$ (5)
 > $\mathrm{Transitivity}\left(\mathrm{TransitiveGroup}\left(\mathrm{op}\left(\mathrm{id}\right)\right)\right)$
 ${4}$ (6)

Find the regular permutation groups of degree 6.

 > $\mathrm{SearchTransitiveGroups}\left(\mathrm{degree}=6,\mathrm{isregular},\mathrm{form}=\mathrm{permgroup}\right)$
 $⟨\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}\right)⟩{,}⟨\left({1}{,}{3}{,}{5}\right)\left({2}{,}{4}{,}{6}\right){,}\left({1}{,}{4}\right)\left({2}{,}{3}\right)\left({5}{,}{6}\right)⟩$ (7)

How many primitive groups are in the transitive groups database.

 > $\mathrm{SearchTransitiveGroups}\left(\mathrm{isprimitive},\mathrm{form}=\mathrm{count}\right)$
 ${288}$ (8)

Note that there are two ways to count the transitive groups of a given degree.

 > $\mathrm{SearchTransitiveGroups}\left(\mathrm{degree}=24,\mathrm{form}=\mathrm{count}\right)=\mathrm{NumTransitiveGroups}\left(24\right)$
 ${25000}{=}{25000}$ (9)

Compatibility

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