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GroupTheory

 SearchPerfectGroups
 search for perfect groups satisfying specified properties

 Calling Sequence SearchPerfectGroups(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", "fpgroup" or "count".

Description

 • The SearchPerfectGroups(spec) command searches Maple's database of perfect groups for groups satisfying properties specified in a sequence spec of search parameters. The valid search parameters may be grouped into several classes, as follows.
 • Use the form = X option to control the form of the output from this command. By default, an expression sequence of IDs for the PerfectGroups database is returned. This is the same as specifying form = "id". To have an expression sequence of groups, either permutation groups, or finitely presented groups, use either the form = "permgroup" or form = "fpgroup" options, respectively. Finally, the form = "count" option causes SearchPerfectGroups 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 PerfectGroups database.  These may differ from the IDs for the same group if it happens to be present in another database, such as the SmallGroups database, which has its own set of group IDs.

Boolean Search Parameters

 • Boolean search parameters p, such as simple, 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 SearchPerfectGroups command to return groups that do, and do not, satisfy the corresponding predicate.
 • The supported boolean search parameters are described in the following table.

 simple describes the class of simple groups quasisimple describes the class of quasi-simple groups indecomposable describes the class of directly indecomposable groups frobenius describes the class of Frobenius groups perfectorderclasses describes the class of groups with perfect order classes

Numeric Search Parameters

 • Maple supports search parameters that describe numeric invariants of finite groups. All have positive integral values. 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. In addition, inequalities of the form p < n (p > n) or p <= n (p >= n) are supported.
 • The supported numeric search parameters are listed in the following table.

 order indicates the order (cardinality) of the group classnumber indicates the number of conjugacy classes of the group compositionlength indicates the number of composition factors of the group degree indicates the degree of the permutation representation stored in the perfect groups database elementordersum indicates the sum of the orders of the elements of the group maxelementorder indicates the largest among the orders of the elements of the group

Subgroup Group Search Parameters

 • A subgroup of a perfect group is typically not perfect.  Therefore, subgroups of perfect groups are indicated by using their ID from the database of small groups.
 • Several subgroup search parameters are supported. These describe the isomorphism type of various subgroups of a group by specifying the Small Group ID (as returned by the IdentifySmallGroup command).
 • For a subgroup or quotient search parameter p, passing an equation of the form p = [ord,id] causes the SearchSmallGroups command to return only groups whose subgroup, or quotient group, corresponding to p are isomorphic to the small group ord/id to be returned.  Passing an equation of the form p = ord causes the SearchSmallGroups command to return only groups whose subgroup, or quotient group, corresponding to p have order ord.
 • The following table describes the supported subgroup search parameters.

 center specifies the SmallGroup ID (or order) of the center of a group socle specifies the SmallGroup ID (or order) of the socle of a group fittingsubgroup specifies the SmallGroup ID (or order) of the Fitting subgroup of a group

 • It is important to understand that the option values for subgroups are the IDs within the small groups database, while the IDs returned by the SearchPerfectGroups command are the IDs of groups within the perfect groups database.

Quotient Group Search Parameters

 • Quotient groups of perfect groups are perfect so, unlike subgroups, quotient groups are specified by their IDs in the perfect groups database.

 centralquotient specifies the PerfectGroup ID (or order) of the central quotient of a group

Examples

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

Count the total number of groups in the database.

 > $\mathrm{SearchPerfectGroups}\left('\mathrm{form}'="count"\right)$
 ${1097}$ (1)

Find the quasisimple perfect groups up to order $500$ that are not simple.

 > $\mathrm{SearchPerfectGroups}\left(\mathrm{order}=1..500,'\mathrm{simple}'=\mathrm{false},'\mathrm{quasisimple}'\right)$
 $\left[{120}{,}{1}\right]{,}\left[{336}{,}{1}\right]$ (2)

Find the quasisimple perfect groups up to order $500$ that are not simple, returned as finitely presented groups.

 > $\mathrm{SearchPerfectGroups}\left(\mathrm{order}=1..500,'\mathrm{simple}'=\mathrm{false},'\mathrm{quasisimple}','\mathrm{form}'="fpgroup"\right)$
 $⟨{}{\mathrm{a1}}{,}{\mathrm{a2}}{,}{\mathrm{a3}}{}{\mid }{}{{\mathrm{a3}}}^{{2}}{,}{{\mathrm{a1}}}^{{2}}{}{{\mathrm{a3}}}^{{-1}}{,}{{\mathrm{a2}}}^{{3}}{,}{{\mathrm{a3}}}^{{-1}}{}{{\mathrm{a2}}}^{{-1}}{}{\mathrm{a3}}{}{\mathrm{a2}}{,}{\mathrm{a1}}{}{\mathrm{a2}}{}{\mathrm{a1}}{}{\mathrm{a2}}{}{\mathrm{a1}}{}{\mathrm{a2}}{}{\mathrm{a1}}{}{\mathrm{a2}}{}{\mathrm{a1}}{}{\mathrm{a2}}{}⟩{,}⟨{}{\mathrm{a1}}{,}{\mathrm{a2}}{,}{\mathrm{a3}}{}{\mid }{}{{\mathrm{a3}}}^{{2}}{,}{{\mathrm{a1}}}^{{2}}{}{{\mathrm{a3}}}^{{-1}}{,}{{\mathrm{a2}}}^{{3}}{,}{{\mathrm{a3}}}^{{-1}}{}{{\mathrm{a2}}}^{{-1}}{}{\mathrm{a3}}{}{\mathrm{a2}}{,}{\mathrm{a1}}{}{\mathrm{a2}}{}{\mathrm{a1}}{}{\mathrm{a2}}{}{\mathrm{a1}}{}{\mathrm{a2}}{}{\mathrm{a1}}{}{\mathrm{a2}}{}{\mathrm{a1}}{}{\mathrm{a2}}{}{\mathrm{a1}}{}{\mathrm{a2}}{}{\mathrm{a1}}{}{\mathrm{a2}}{,}{{\mathrm{a1}}}^{{-1}}{}{{\mathrm{a2}}}^{{-1}}{}{\mathrm{a1}}{}{\mathrm{a2}}{}{{\mathrm{a1}}}^{{-1}}{}{{\mathrm{a2}}}^{{-1}}{}{\mathrm{a1}}{}{\mathrm{a2}}{}{{\mathrm{a1}}}^{{-1}}{}{{\mathrm{a2}}}^{{-1}}{}{\mathrm{a1}}{}{\mathrm{a2}}{}{{\mathrm{a1}}}^{{-1}}{}{{\mathrm{a2}}}^{{-1}}{}{\mathrm{a1}}{}{\mathrm{a2}}{}{{\mathrm{a3}}}^{{-1}}{}⟩$ (3)

Find the quasisimple perfect groups up to order $500$ that are not simple, returned as permutation groups.

 > $\mathrm{SearchPerfectGroups}\left(\mathrm{order}=1..500,'\mathrm{simple}'=\mathrm{false},'\mathrm{quasisimple}','\mathrm{form}'="permgroup"\right)$
 $⟨\left({1}{,}{2}{,}{5}{,}{3}\right)\left({4}{,}{7}{,}{6}{,}{8}\right)\left({9}{,}{13}{,}{11}{,}{14}\right)\left({10}{,}{15}{,}{12}{,}{16}\right)\left({17}{,}{19}{,}{18}{,}{20}\right)\left({21}{,}{24}{,}{23}{,}{22}\right){,}\left({1}{,}{4}{,}{2}\right)\left({3}{,}{5}{,}{6}\right)\left({7}{,}{9}{,}{10}\right)\left({8}{,}{11}{,}{12}\right)\left({13}{,}{16}{,}{17}\right)\left({14}{,}{15}{,}{18}\right)\left({19}{,}{21}{,}{22}\right)\left({20}{,}{23}{,}{24}\right){,}\left({1}{,}{5}\right)\left({2}{,}{3}\right)\left({4}{,}{6}\right)\left({7}{,}{8}\right)\left({9}{,}{11}\right)\left({10}{,}{12}\right)\left({13}{,}{14}\right)\left({15}{,}{16}\right)\left({17}{,}{18}\right)\left({19}{,}{20}\right)\left({21}{,}{23}\right)\left({22}{,}{24}\right)⟩{,}⟨\left({1}{,}{2}{,}{5}{,}{3}\right)\left({4}{,}{7}{,}{6}{,}{8}\right)\left({9}{,}{13}{,}{11}{,}{14}\right)\left({10}{,}{15}{,}{12}{,}{16}\right){,}\left({1}{,}{4}{,}{2}\right)\left({3}{,}{5}{,}{6}\right)\left({7}{,}{9}{,}{10}\right)\left({8}{,}{11}{,}{12}\right){,}\left({1}{,}{5}\right)\left({2}{,}{3}\right)\left({4}{,}{6}\right)\left({7}{,}{8}\right)\left({9}{,}{11}\right)\left({10}{,}{12}\right)\left({13}{,}{14}\right)\left({15}{,}{16}\right)⟩$ (4)

Count the number of non-simple, quasisimple perfect groups in the database.

 > $\mathrm{SearchPerfectGroups}\left('\mathrm{simple}'=\mathrm{false},'\mathrm{quasisimple}','\mathrm{form}'="count"\right)$
 ${54}$ (5)

Find the quasi-simple perfect groups in the databaswe whose centre has order equal to $4$.

 > $\mathrm{SearchPerfectGroups}\left('\mathrm{quasisimple}','\mathrm{centre}'=4\right)$
 $\left[{80640}{,}{2}\right]{,}\left[{80640}{,}{3}\right]{,}\left[{80640}{,}{4}\right]{,}\left[{116480}{,}{1}\right]$ (6)

Find the quasi-simple perfect groups in the databaswe whose centre is isomorphic to SmallGroup( 4,2 ).

 > $\mathrm{SearchPerfectGroups}\left('\mathrm{quasisimple}','\mathrm{centre}'=\left[4,2\right]\right)$
 $\left[{80640}{,}{2}\right]{,}\left[{116480}{,}{1}\right]$ (7)
 > $\mathrm{IdentifySmallGroup}\left(\mathrm{Centre}\left(\mathrm{PerfectGroup}\left(80640,2\right)\right)\right)$
 ${4}{,}{2}$ (8)

Take care to recognise the difference between the ID of a group in the perfect groups database and the (normally different!) ID in the small groups database when a group is present in both databases.

For example, the alternating group of degree $5$ has ID [ 60, 5 ] in the database of small groups (it is the fifth group of order $60$) , but ID [60, 1] in the perfect groups database (it is the first perfect group of order $60$).

 > $\mathrm{IdentifySmallGroup}\left(\mathrm{Alt}\left(5\right)\right)$
 ${60}{,}{5}$ (9)
 > $\mathrm{SearchPerfectGroups}\left(\mathrm{socle}=\left[60,5\right]\right)$
 $\left[{60}{,}{1}\right]$ (10)
 > $\mathrm{IdentifySmallGroup}\left(\mathrm{PerfectGroup}\left(60,1\right)\right)$
 ${60}{,}{5}$ (11)

Compatibility

 • The GroupTheory[SearchPerfectGroups] command was introduced in Maple 2019.