search for Frobenius groups satisfying specified properties
SearchFrobeniusGroups(spec, output, formopt )
expression sequence of search parameters
(optional) an option of the form output = "list" (the default), output = "iterator", or output="count"
(optional) an option of the form form = X, where X is one of "id" (the default) or "permgroup"
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.
describes groups with an Abelian Frobenius complement
describes groups with an Abelian Frobenius kernel
describes groups all of whose Sylow subgroups are abelian
describes the class of complete groups
describes groups with a cyclic Frobenius complement
describes groups with a cyclic Frobenius kernel
describes groups whose Sylow subgroups are all cyclic
describes groups with an elementary abelian Frobenius kernel
describes groups with a homocyclic Frobenius kernel
describes Lagrangian groups, that is, those satisfying the converse of Lagrange's Theorem
describes the class of metabelian groups
describes the class of metcyclic groups
describes groups with a nilpotent Frobenius complement
describes the class of groups with an ordered Sylow tower
describes the class of perfect groups
describes the class of groups with perfect order classes
describes the class of primitive groups
describes the class of soluble groups
describes the class of supersoluble groups
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.
indicates the number of conjugacy classes of the group
indicates the number of order classes of the group
indicates the sum of the orders of the elements of the group
indicates the largest order of an element of the group
indicates the number of composition factors of the group
indicates the length of the lower Fitting series of the group
indicates the length of the Frattini series of the group
indicates the exponent of the group
indicates the minimum number of generators
indicates the nilpotency class of the Frobenius kernel
indicates the order (cardinality) of the group
indicates the permutation group rank (number of sub-orbits) of the group
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.
specifies the SmallGroup ID (or order) of the Frobenius complement
specifies the SmallGroup ID (or order) of the derived subgroup
specifies the SmallGroup ID (or order) of the derived quotient
specifies the SmallGroup ID (or order) of the Frobenius kernel
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.
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.
What are the Frobenius groups of order 100?
Let's check that each of these has an abelian Frobenius kernel.
However, not all have a cyclic Frobenius kernel.
Let's see what other Frobenius groups have a cyclic kernel of order 25.
We get the same result if we further specify that the Frobenius complement be nilpotent.
The following command counts the number of Frobenius groups of order at most 1000 with Frobenius complement of order 4.
Find the doubly transitive Frobenius groups in the database with a homocyclic Frobenius kernel and order greater than 10000.
Which Frobenius groups in the database are metabelian but not metacyclic and have order at least 100 but less than 300?
Count the Frobenius groups with rank equal to 7.
Count the number of (just) singly transitive Frobenius groups in the database.
We can construct an iterator for the output, and use it to search for a primitive group, stopping at the first example.
it ≔ SearchFrobeniusGroups⁡'transitivity'=1,'output'=iterator,'form'=permgroup
it≔⟨Iterator for Frobenius Groups Query: "transitivity = 1", with 8984 results⟩
forid,GinitdoifIsPrimitive⁡Gthenprint⁡id;breakend ifend do:
The GroupTheory[SearchFrobeniusGroups] command was introduced in Maple 2019.
For more information on Maple 2019 changes, see Updates in Maple 2019.
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