Special ag groups are a subcategory of ag groups (see Finite Polycyclic Groups).
Let G be an ag group with PAG system (g_{1}, ..., g_{n}). Then (g_{1}, ..., g_{n}) is a special ag system if it is an ag system with some additional properties, which are described below.
In general a finite polycyclic group has several different ag systems and at least one of this is a special ag system, but in GAP3 an ag group is defined by a fixed ag system and according to this an ag group is called a special ag group if its ag system is a special ag system.
Special ag systems give more information about their corresponding group than arbitrary ag systems do (see More about Special Ag Groups) and furthermore there are many algorithms, which are much more efficient for a special ag group than for an arbitrary one. (See Ag Group Functions for Special Ag Groups)
The following sections describe the special ag system (see More about Special Ag Groups), their construction in GAP3 (see Construction of Special Ag Groups and Restricted Special Ag Groups) and their additional record entries (see Special Ag Group Records). Then follow two sections with functions which do only work for special ag groups (see MatGroupSagGroup and DualMatGroupSagGroup).
Now the properties of a special ag system are described. First of all the Leedham-Green series will be introduced.
Let G = G_{1} > G_{2} > ... > G_{m} > G_{m+1} = { 1 } be the lower nilpotent series of G, i.e., G_{i} is the smallest normal subgroup of G_{i-1} such that G_{i-1} / G_{i} is nilpotent.
To refine this series the lower elementary abelian series of a nilpotent group N will be constructed. Let N = P_{1} . ... . P_{l} be the direct product of its Sylow-subgroups such that P_{h} is a p_{h}-group and p_{1} < p_{2} < ... < p_{l} holds. Let λ_{j}(P_{h}) be the j-th term of the p_{h}-central series of P_{h} and let k_{h} be the length of this series (see PCentralSeries). Define N_{j, ph} as the subgroup of N with
N_{j, ph} = λ_{j+1}(P_{1}) ... λ_{j+1}(P_{h-1}) . λ_{j}(P_{h}) ... λ_{j}(P_{l}). |
N = N_{1, p1} ≥ N_{1, p2} ≥ ... ≥ N_{1,pl} ≥ N_{2, p1} ≥ ... ≥ N_{k, pl} = { 1 } |
To get the Leedham-Green series of G, each factor of the lower nilpotent series of G is refined by its lower elementary abelian series. The subgroups of the Leedham-Green series are denoted by G_{i, j, pi, h} such that G_{i, j, pi, h} / G_{i+1} = (G_{i} / G_{i+1})_{j, pi,h} for each prime p_{i,h} dividing the order of G_{i} / G_{i+1}. The Leedham-Green series is a characteristic series with elementary abelian factors.
A PAG system corresponds naturally to a composition series of its group. The first additional property of a special ag system is that the corresponding composition series refines the Leedham-Green series.
Secondly, all the elements of a special ag system are of prime-power order, and furthermore, if a set of primes π = {q_{1}, ..., q_{r}} is given, all elements of a special ag system which are of q_{h}-power order for some q_{h} in π generate a Hall-π-subgroup of G. In fact they form a canonical generating sequence of the Hall-π-subgroup. These Hall subgroups are called public subgroups, since a subset of the PAG system is an induced generating set for the subgroup. Note that the set of all public Sylow subgroups forms a Sylow system of G.
The last property of the special ag systems is the existence of public local head complements. For a nilpotent group N, the group
λ_{2}(N) = λ_{2}(P_{1}) ... λ_{2}(P_{l}) |
(G_{i} / G_{i+1}) / λ_{2}(G_{i} / G_{i+1}) = G_{i} / G_{i, 2,pi,1} |
λ_{2}(G_{i} / G_{i+1}) = G_{i, 2,pi,1} / G_{i+1} |
To handle the special ag system the weights are introduced. Let (g_{1}, ..., g_{n}) be a special ag system. The triple (w_{1}, w_{2}, w_{3}) is called the weight of the generator g_{i} if g_{i} lies in G_{w1, w2, w3} but not lower down in the Leedham-Green series. That means w_{1} corresponds to the subgroup in the lower nilpotent series and w_{2} to the subgroup in the elementary-abelian series of this factor, and w_{3} is the prime dividing the order of g_{i}. Then weight(g_{i}) = (w_{1}, w_{2}, w_{3}) and weight_{j}(g_{i}) = w_{j} for j = 1,2,3 is set. With this definition {g_{i} | weight_{3}(g_{i}) ∈ π} is a Hall-π-subgroup of G and {g_{i} | weight(g_{i}) ≠ (j, 1, p) for some p } is a local head complement.
Now some advantages of a special ag system are summarized.
•[1.] You have a characteristic series with elementary abelian factors of G explicitly given in the ag system. This series is refined by the composition series corresponding to the ag system.
•[2.] You can see whether G is nilpotent or even a p-group, and if it is, you have a central series explicitly given by the Leedham-Green series. Analogously you can see whether the group is even elementary abelian.
•[3.] You can easily calculate Hall-π-subgroups of G. Furthermore the set of public Sylow subgroups forms a Sylow system.
•[4.] You get a smaller generating set of the group by taking only the elements which correspond to local heads of the group.
•[5.] The collection with a special ag system may be faster than the collection with an arbitrary ag system, since in the calculation of the public subgroups of G the commutators of the ag generators are shortened.
•[6.] Many algorithms are faster for special ag groups than for arbitrary ag groups.
SpecialAgGroup( G )
The function SpecialAgGroup
takes an ag group G as input and
calculates a special ag group H, which is isomorphic to G.
To obtain the additional information of a special ag system see Special Ag Group Records.
If one is only interested in some of the information of special ag
systems then it is possible to suppress the calculation of one or all
types of the public subgroups by calling the function
SpecialAgGroup( G, flag )
, where flag is "noHall", "noHead" or
"noPublic".
With this options the algorithm takes less time. It calculates an ag
group H, which is isomorphic to G. But be careful, because the output
H is not handled as a special ag group by GAP3 but as an arbitrary ag
group. Exspecially none of the functions listet in Ag Group Functions
for Special Ag Groups use the algorithms for special ag groups.
SpecialAgGroup( G, "noPublic" )
calculates an ag group H, which is isomorphic to G and whose ag system is corresponding to the Leedham-Green series.
SpecialAgGroup( G, "noHall" )
calculates an ag group H, which is isomorphic to G and whose ag system is corresponding to the Leedham-Green series and has public local head complements.
SpecialAgGroup( G, "noHead" )
calculates an ag group H, which is isomorphic to G and whose ag system is corresponding to the Leedham-Green series and has public Hall subgroups.
To obtain the additional information of a special ag system see Special Ag Group Records.
In addition to the record components of ag groups (see Finite Polycyclic Groups) the following components are present in the group record of a special ag group H.
weights
:
The entries layers
, first
, head
and tail
only depend on the
weights
. These entries are useful in many of the programs using the
special ag system.
layers
:weights
.
first
:first
[j] = i if h_{i} is the
first element of the j-th layer. Additionally the last entry of
the list first
is always n + 1.
head
:head
[j] = i if h_{i} is the
first element of the j-th local head. Additionally the last
entry of the list head
is always n + 1 (see More about Special
Ag Groups).
tail
:tail
[j] = i if h_{i-1} is
the last element of the j-th local head. In other words h_{i} is either
the first element of the tail of the j-th layer in the lower nilpotent
series, or in case this tail is trivial, then h_{i} is the first
element of the j+1-st layer in the lower nilpotent series.
If the tail of the smallest nontrivial subgroup of the lower nilpotent
series is trivial, then the last entry of the list tail
is n+1
(see More about Special Ag Groups).
bijection
:The next four entries indicate if any flag and which one is used in the calculation of the special ag system (see Construction of Special Ag Groups and Restricted Special Ag Groups).
isHallSystem
:
isHeadSystem
:
isSagGroup
:
Note that in GAP3 an ag group is called a special ag group if and only
if the record entry isSagGroup
is true.
# construct a wreath product of a4 with s3 where s3 operates on 3 points. gap> s3 := SymmetricGroup( AgWords, 3 );; gap> a4 := AlternatingGroup( AgWords, 4 );; gap> a4wrs3 := WreathProduct(a4, s3, s3.bijection); Group( h1, h2, n1_1, n1_2, n1_3, n2_1, n2_2, n2_3, n3_1, n3_2, n3_3 ) # now calculate the special ag group gap> S := SpecialAgGroup( a4wrs3 ); Group( h1, n3_1, h2, n2_1, n1_1, n1_2, n1_3, n2_2, n2_3, n3_2, n3_3 ) gap> S.weights; [ [ 1, 1, 2 ], [ 1, 1, 3 ], [ 2, 1, 3 ], [ 2, 1, 3 ], [ 2, 2, 3 ], [ 3, 1, 2 ], [ 3, 1, 2 ], [ 3, 1, 2 ], [ 3, 1, 2 ], [ 3, 1, 2 ], [ 3, 1, 2 ] ] gap> S.layers; [ 1, 2, 3, 3, 4, 5, 5, 5, 5, 5, 5 ] gap> S.first; [ 1, 2, 3, 5, 6, 12 ] gap> S.head; [ 1, 3, 6, 12 ] gap> S.tail; [ 3, 5, 12 ] gap> S.bijection; GroupHomomorphismByImages( Group( h1, n3_1, h2, n2_1, n1_1, n1_2, n1_3, n2_2, n2_3, n3_2, n3_3 ), Group( h1, h2, n1_1, n1_2, n1_3, n2_1, n2_2, n2_3, n3_1, n3_2, n3_3 ), [ h1, n3_1, h2, n2_1, n1_1, n1_2, n1_3, n2_2, n2_3, n3_2, n3_3 ], [ h1, n3_1, h2, n2_1*n3_1^2, n1_1*n2_1*n3_1, n1_2, n1_3, n2_2, n2_3, n3_2, n3_3 ] ) gap> S.isHallSystem; true gap> S.isHeadSystem; true gap> S.isSagGroup; true
In the next sections the functions which only apply to special ag groups are described.
MatGroupSagGroup( H, i )
MatGroupSagGroup
calculates the matrix representation of H on the
i-th layer of the Leedham-Green series of H (see More about
Special Ag Groups).
See also MatGroupAgGroup
.
gap> S := SpecialAgGroup( a4wrs3 );; gap> S.weights; [ [ 1, 1, 2 ], [ 1, 1, 3 ], [ 2, 1, 3 ], [ 2, 1, 3 ], [ 2, 2, 3 ], [ 3, 1, 2 ], [ 3, 1, 2 ], [ 3, 1, 2 ], [ 3, 1, 2 ], [ 3, 1, 2 ], [ 3, 1, 2 ] ] gap> MatGroupSagGroup(S,3); Group( [ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3) ] ], [ [ Z(3)^0, Z(3)^0 ], [ 0*Z(3), Z(3)^0 ] ] )
DualMatGroupSagGroup( H, i )
DualMatGroupSagGroup
calculates the dual matrix representation of H
on the i-th layer of the Leedham-Green series of H (see More about
Special Ag Groups).
Let V be an F H-module for a field F. Then the dual module to V is defined by V^{*} := {f : V → F | f is linear }. This module is also an F H-module and the dual matrix representation is the representation on the dual module.
gap> S := SpecialAgGroup( a4wrs3 );; gap> DualMatGroupSagGroup(S,3); Group( [ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3) ] ], [ [ Z(3)^0, 0*Z(3) ], [ Z(3)^0, Z(3)^0 ] ] )
Since special ag groups are ag groups all functions for ag groups are applicable to special ag groups. However certain of these functions use special implementations to treat special ag groups, i.e. there exists functions like SagGroupOps.FunctionName, which are called by the corresponding general function in case a special ag group given. If you call one of these general functions with an arbitrary ag group, the general function will not calculate the special ag group but use the function for ag groups. For the special implementations to treat special ag groups note the following.
Centre( H )
MinimalGeneratingSet( H )
Intersection( U, L)
EulerianFunction( H )
MaximalSubgroups( H )
ConjugacyClassesMaximalSubgroups( H )
PrefrattiniSubgroup( H )
FrattiniSubgroup( H )
IsNilpotent( H )
These functions are often faster and often use less space for special ag
groups.
ElementaryAbelianSeries( H )
This function returns the Leedham-Green series (see More about Special
Ag Groups).
IsElementaryAbelianSeries( H )
Returns true.
HallSubgroup( H, primes )
SylowSubgroup( H, p )
SylowSystem( H )
These functions return the corresponding public subgroups (see More
about Special Ag Groups).
Subgroup( H, gens )
AgSubgroup( H, gens, bool )
These functions return an ag group which is not special, except if the
group itself is returned.
All domain functions not mentioned here use no special treatments for
special ag groups.
Note also that there exists a package to compute formation theoretic
subgroups of special ag groups. This may be used to compute the
system normalizer of the public Sylow system, which is the F-normalizer
for the formation of nilpotent groups F. It is also possible to
compute F-normalizers as well as F-covering subgroups and
F-residuals of special ag groups for a number of saturated formations
F which are given within the package or for self-defined saturated
formations F.
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