
Calling Sequence


SearchFrobeniusGroups(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 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 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 groups, either permutation groups, or finitely presented groups, use either the form = "permgroup" or form = "fpgroup" options, respectively. Finally, the form = "count" option causes SearchFrobeniusGroups 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 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.

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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


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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

complete

describes the class of complete groups

cycliccomplement

describes groups with a cyclic Frobenius complement

cyclickernel

describes groups with a cyclic Frobenius kernel

elementarykernel

describes groups with an elementary abelian Frobenius kernel

homocyclickernel

describes groups with a homocyclic Frobenius kernel

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 endpoints) 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

frattinilength

indicates the length of the Frattini series of the group

exponent

indicates the exponent of the group

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 suborbits) of the group

transitivity

indicates the transitivity of the group

nsylow[ p ]

indicates the number of Sylow psubgroups of the group





Subgroup and Quotient Search Parameters


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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.

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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.

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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.

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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 psubgroup



•

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 form = "count" option.
>

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

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.
>

$\mathrm{IsAbelian}\left(\mathrm{FrobeniusKernel}\left(\mathrm{FrobeniusGroup}\left(100\,1\right)\right)\right)$

>

$\mathrm{IsAbelian}\left(\mathrm{FrobeniusKernel}\left(\mathrm{FrobeniusGroup}\left(100\,2\right)\right)\right)$

>

$\mathrm{IsAbelian}\left(\mathrm{FrobeniusKernel}\left(\mathrm{FrobeniusGroup}\left(100\,3\right)\right)\right)$

However, not all have a cyclic Frobenius kernel.
>

$\mathrm{IsCyclic}\left(\mathrm{FrobeniusKernel}\left(\mathrm{FrobeniusGroup}\left(100\,1\right)\right)\right)$

>

$\mathrm{IsCyclic}\left(\mathrm{FrobeniusKernel}\left(\mathrm{FrobeniusGroup}\left(100\,2\right)\right)\right)$

>

$\mathrm{IsCyclic}\left(\mathrm{FrobeniusKernel}\left(\mathrm{FrobeniusGroup}\left(100\,3\right)\right)\right)$

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 $4$.
>

$\mathrm{SearchFrobeniusGroups}\left('\mathrm{order}'\le 1000\,'\mathrm{complement}'=4\,'\mathrm{form}'=''count''\right)$

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

$\mathrm{SearchFrobeniusGroups}\left('\mathrm{rank}'=7\,'\mathrm{form}'=''count''\right)$



Compatibility


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

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The GroupTheory[SearchFrobeniusGroups] command was updated in Maple 2021.



