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GroupTheory

 SearchFrobeniusGroups
 search for Frobenius groups satisfying specified properties Calling Sequence SearchFrobeniusGroups(spec, output, formopt ) Parameters

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

 • The SearchFrobeniusGroups( spec ) command searches Maple's database of Frobenius groups for groups satisfying properties specified in a sequence spec of search parameters. The valid search parameters may be grouped into several classes, as described in the following sections.
 • Use the output = X option to control the kind of output returned. By default, an expression sequence of results, corresponding to the output = "list" option is produced. Alternatively, you can request a simple count of the results by using the output = "count" option. In the event that the number of returned results is large, the output = "iterator" option is useful, as it returns an iterator object that you can use to iterate over the results one at a time.
 • Use the form = X option to control the form of the output from this command. By default, an expression sequence of IDs for the FrobeniusGroups 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 option has no meaning if the output = "count" option is specified.
 • Note that the IDs returned in the default case are the IDs of the groups within the FrobeniusGroups 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.
 • Note further that IDs returned by SearchFrobeniusGroups are limited to those actually present in the database. In particular, they are limited by the maximum group order and by the order exclusions documented in FrobeniusGroup. Boolean Search Parameters

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

 abeliancomplement describes groups with an Abelian Frobenius complement abeliankernel describes groups with an Abelian Frobenius kernel abeliansylow describes groups all of whose Sylow subgroups are abelian complete describes the class of complete  groups cycliccomplement describes groups with a cyclic Frobenius complement cyclickernel describes groups with a cyclic Frobenius kernel cyclicSylow describes groups whose Sylow subgroups are all cyclic elementarykernel describes groups with an elementary abelian Frobenius kernel homocyclickernel describes groups with a homocyclic Frobenius kernel lagrangian describes Lagrangian groups, that is, those satisfying the converse of Lagrange's Theorem metabelian describes the class of metabelian groups metacyclic describes the class of metcyclic groups nilpotentcomplement describes groups with a nilpotent Frobenius complement orderedsylowtower describes the class of groups with an ordered Sylow tower perfect describes the class of perfect groups perfectorderclasses describes the class of groups with perfect order classes primitive describes the class of primitive groups soluble describes the class of soluble groups supersoluble describes the class of supersoluble groups sylowtower describes the class of groups with a Sylow tower (of any complexion) 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 indicated 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.

 classnumber indicates the number of conjugacy classes of the group orderclassnumber indicates the number of order classes of the group elementordersum indicates the sum of the orders of the elements of the group maxelementorder indicates the largest order of an element of the group derivedlength indicates the number of composition factors of the group fittinglength indicates the length of the lower Fitting series of the group frattinilength indicates the length of the Frattini series of the group exponent indicates the exponent of the group generationrank indicates the minimum number of generators kernel_nilpclass indicates the nilpotency class of the Frobenius kernel order indicates the order (cardinality) of the group rank indicates the permutation group rank (number of sub-orbits) of the group transitivity indicates the transitivity of the group nsylow[ p ] indicates the number of Sylow p-subgroups of the group Subgroup and Quotient Search Parameters

 • A subgroup of a Frobenius group is typically not a Frobenius group. (It may be in some cases, of course.) Therefore, subgroups of Frobenius groups are indicated by using their ID from the database of small groups. In some cases, only the order of the subgroup is stored, since the subgroup is larger than any group in the SmallGroups database.
 • 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), or just the order of the group if is too large to have a SmallGroups database ID.
 • 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.

 complement specifies the SmallGroup ID (or order) of the Frobenius complement derivedsubgroup specifies the SmallGroup ID (or order) of the derived subgroup derivedquotient specifies the SmallGroup ID (or order) of the derived quotient kernel specifies the SmallGroup ID (or order) of the Frobenius kernel sylow[ p] specifies the SmallGroup ID (or order) of the Sylow p-subgroup

 • 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 SearchFrobeniusGroups command are the IDs of groups within the FrobeniusGroups database. Examples

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

The following command places no restrictions on the groups being queried, so it just returns the total number of Frobenius groups in the database because of the output = "count" option.

 > $\mathrm{SearchFrobeniusGroups}\left('\mathrm{output}'="count"\right)$
 ${9034}$ (1)

What are the Frobenius groups of order $100$?

 > $\mathrm{SearchFrobeniusGroups}\left('\mathrm{order}'=100\right)$
 $\left[{100}{,}{1}\right]{,}\left[{100}{,}{2}\right]{,}\left[{100}{,}{3}\right]$ (2)

Let's check that each of these has an abelian Frobenius kernel.

 > $\mathrm{IsAbelian}\left(\mathrm{FrobeniusKernel}\left(\mathrm{FrobeniusGroup}\left(100,1\right)\right)\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{IsAbelian}\left(\mathrm{FrobeniusKernel}\left(\mathrm{FrobeniusGroup}\left(100,2\right)\right)\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{IsAbelian}\left(\mathrm{FrobeniusKernel}\left(\mathrm{FrobeniusGroup}\left(100,3\right)\right)\right)$
 ${\mathrm{true}}$ (5)

However, not all have a cyclic Frobenius kernel.

 > $\mathrm{IsCyclic}\left(\mathrm{FrobeniusKernel}\left(\mathrm{FrobeniusGroup}\left(100,1\right)\right)\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{IsCyclic}\left(\mathrm{FrobeniusKernel}\left(\mathrm{FrobeniusGroup}\left(100,2\right)\right)\right)$
 ${\mathrm{false}}$ (7)
 > $\mathrm{IsCyclic}\left(\mathrm{FrobeniusKernel}\left(\mathrm{FrobeniusGroup}\left(100,3\right)\right)\right)$
 ${\mathrm{false}}$ (8)

Let's see what other Frobenius groups have a cyclic kernel of order $25$.

 > $\mathrm{SearchFrobeniusGroups}\left(\mathrm{kernel}=25,'\mathrm{cyclickernel}'\right)$
 $\left[{50}{,}{1}\right]{,}\left[{100}{,}{1}\right]$ (9)

We get the same result if we further specify that the Frobenius complement be nilpotent.

 > $\mathrm{SearchFrobeniusGroups}\left(\mathrm{kernel}=25,'\mathrm{cyclickernel}','\mathrm{nilpotentcomplement}'\right)$
 $\left[{50}{,}{1}\right]{,}\left[{100}{,}{1}\right]$ (10)

The following command counts the number of Frobenius groups of order at most $1000$ with Frobenius complement of order $4$.

 > $\mathrm{SearchFrobeniusGroups}\left('\mathrm{order}'\le 1000,'\mathrm{complement}'=4,'\mathrm{output}'="count"\right)$
 ${59}$ (11)

Find the doubly transitive Frobenius groups in the database with a homocyclic Frobenius kernel and order greater than $10000$.

 > $\mathrm{SearchFrobeniusGroups}\left(10000<'\mathrm{order}',1<'\mathrm{transitivity}','\mathrm{homocyclickernel}'\right)$
 $\left[{10100}{,}{1}\right]{,}\left[{10506}{,}{1}\right]{,}\left[{11342}{,}{1}\right]{,}\left[{11772}{,}{1}\right]{,}\left[{12656}{,}{1}\right]{,}\left[{14520}{,}{1}\right]{,}\left[{14520}{,}{2}\right]{,}\left[{14520}{,}{3}\right]{,}\left[{14520}{,}{4}\right]$ (12)

Which Frobenius groups in the database are metabelian but not metacyclic and have order at least $100$ but less than $300$?

 > $\mathrm{SearchFrobeniusGroups}\left(100\le '\mathrm{order}','\mathrm{order}'<300,'\mathrm{metabelian}','\mathrm{metacyclic}'=\mathrm{false}\right)$
 $\left[{100}{,}{2}\right]{,}\left[{100}{,}{3}\right]{,}\left[{126}{,}{2}\right]{,}\left[{147}{,}{2}\right]{,}\left[{147}{,}{3}\right]{,}\left[{150}{,}{1}\right]{,}\left[{150}{,}{3}\right]{,}\left[{156}{,}{2}\right]{,}\left[{162}{,}{2}\right]{,}\left[{162}{,}{3}\right]{,}\left[{162}{,}{4}\right]{,}\left[{162}{,}{5}\right]{,}\left[{180}{,}{1}\right]{,}\left[{192}{,}{1}\right]{,}\left[{192}{,}{2}\right]{,}\left[{192}{,}{5}\right]{,}\left[{196}{,}{1}\right]{,}\left[{198}{,}{2}\right]{,}\left[{200}{,}{1}\right]{,}\left[{228}{,}{1}\right]{,}\left[{234}{,}{2}\right]{,}\left[{240}{,}{1}\right]{,}\left[{242}{,}{2}\right]{,}\left[{250}{,}{2}\right]{,}\left[{250}{,}{3}\right]{,}\left[{270}{,}{2}\right]{,}\left[{270}{,}{3}\right]{,}\left[{294}{,}{2}\right]{,}\left[{294}{,}{3}\right]{,}\left[{294}{,}{5}\right]$ (13)

Count the Frobenius groups with rank equal to $7$.

 > $\mathrm{SearchFrobeniusGroups}\left('\mathrm{rank}'=7,'\mathrm{output}'="count"\right)$
 ${39}$ (14)

Count the number of (just) singly transitive Frobenius groups in the database.

 > $\mathrm{SearchFrobeniusGroups}\left('\mathrm{transitivity}'=1,'\mathrm{output}'="count"\right)$
 ${8984}$ (15)

We can construct an iterator for the output, and use it to search for a primitive group, stopping at the first example.

 > $\mathrm{it}≔\mathrm{SearchFrobeniusGroups}\left('\mathrm{transitivity}'=1,'\mathrm{output}'="iterator",'\mathrm{form}'="permgroup"\right)$
 ${\mathrm{it}}{≔}{\mathrm{⟨Iterator for Frobenius Groups Query: "transitivity = 1", with 8984 results⟩}}$ (16)
 > $\mathbf{for}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{id},G\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{it}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{if}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{IsPrimitive}\left(G\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{then}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{print}\left(\mathrm{id}\right);\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{break}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end if}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end do}:$
 ${10}{,}{1}$ (17) Compatibility

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