Given the matrix group G, construct the subgroup H of G generated by the elements specified by the list L, where L is a list of one or more items of the following types:Each element or group specified by the list must belong to the same generic matrix group. The subgroup H will be constructed as a subgroup of some group which contains each of the elements and groups specified in the list.
- (a)
- A sequence of n integers defining a matrix of G;
- (b)
- A set or sequence of sequences of type (a);
- (c)
- An element of G;
- (d)
- A set or sequence of elements of G;
- (e)
- A subgroup of G;
- (f)
- A set or sequence of subgroups of G.
The generators of H consist of the elements specified by the terms of the list L together with the stored generators for groups specified by terms of the list. Repetitions of an element and occurrences of the identity element are removed.
Given the matrix group G, construct the subgroup H of G that is the normal closure of the subgroup H generated by the elements specified by the list L, where the possibilities for L are the same as for the sub-constructor.
> GL42 := GeneralLinearGroup(4, GF(2)); > Ominus42 := sub< GL42 | [1,0,0,0, 1,1,0,1, 1,0,1,0, 0,0,0,1 ], > [0,1,0,0, 1,0,0,0, 0,0,1,0, 0,0,0,1 ], > [0,1,0,0, 1,0,0,0, 0,0,1,0, 0,0,1,1 ] >; > Order(Ominus42); 120 > H := sub< Ominus42 | $.1, $.3 >; print Order(H); 10 > N := ncl< Ominus42 | $.1, $.3 >; > Order(N); 60
The index of the subgroup H in the group G. The index is returned as an integer. If the orders of G and H are not known, they will be computed.
The index of the subgroup H in the group G. The index is returned as a factored integer. The factorization is returned in the form of a sequence Q which is defined as follows: If [ G : H ] = p1e1 ... pnen, ei != 0, then Q will be the integer sequence [ <p1, e1>, ..., <pn, en> ]. If the orders of G and H are not known, they will be computed.
Returns true if the subgroup H of the group G lies in the centre of G, false otherwise.
Returns true if the subgroup H of the group G is a maximal subgroup of G. This function is evaluated by constructing the permutation representation of G on the cosets of H and testing this representation for primitivity. For this reason, the use of IsMaximal should be avoided if the index of H in G exceeds a few thousand.
Returns true if the subgroup H of the group G is a normal subgroup of G, false otherwise.
Returns true if the subgroup H of the group G is subnormal in G, false otherwise.
Construct the conjugate g - 1 * H * g of the matrix group H by the matrix g. The group H and the element g must belong to a common matrix group.
Given groups H and K which belong to the same matrix group, construct the intersection of H and K.
Given subgroups H and K of the group G, construct the commutator subgroup of H and K as a subgroup of G. If K is a subgroup of H, then G may be omitted.
Construct the centralizer of the matrix g in the group G; g and G must belong to a common matrix group.
Construct the centralizer of the group H in the group G; G and H must belong to a common matrix group.
Given a subgroup H of the matrix group G, construct the maximal normal subgroup of G that is contained in the subgroup H.
Given a subgroup H of the matrix group G, construct the normal closure of H in G.
Given a subgroup H of the group G, construct the normalizer of H in G.
Print: RngIntElt Default: 0
OverGroup: RngIntElt Default: 0
Given H ≤GLn(q), return the normalizer of H in GLn(q). The algorithm is described in Coutts [Cou11]. It exploits Aschbacher's description of the maximal subgroups of a classical group to obtain an overgroup M of the normaliser N and, where necessary, then applies LMGNormaliser (M, H) to construct N. If the optional argument OverGroup is true, then return M, so skipping the last step. The optional argument Print takes values between 0 and 2, and determines the amount of information printed.
Given a group G and a prime p, construct the Sylow p-subgroup of G.
Given a group G and a prime p dividing the order of G, construct the maximal normal p-subgroup of G.
Returns a sequence of subgroups of G, each with index at most n. The sequence will contain one representative from each conjugacy class of G-subgroups satisfying the index constraint. The algorithm used is described in Cannon, Holt, Slattery & Steel [CHSS03].The previous version of the algorithm is available by setting the parameter Algorithm to the string "Subgroups". In this case the group G is subject to the same restrictions as the group input to the Subgroups function above.
In the second form t should be a pair of integers < a, b >, and subgroups with index in the interval [a, b] will be returned.
Other parameters are Presentation which may be set true to return a second sequence of presentations of the groups found, and Print which may be set to a positive integer to turn on diagnostic printing of the progress of the algorithms.
Same as above, but only those subgroups containing the normal subgroup N of G are returned. This is equivalent to computing the low index subgroups of G/N. Only the Print parameter is available for this command.
Given a matrix group G, and an expression R defining a positive integer range (see below), determine the conjugacy classes of subgroups of G whose indices lie in the range specified by R. The subgroups are returned as a sequence of subgroups of G. The argument R is one of the following:This is an alternative method for computing low index subgroups that uses an algorithm due to Leedham-Green & O'Brien [LGO02]. In practice, the algorithm is most useful for small values of n, say up to 8.
- (a)
- An integer n representing the range [1, n];
- (b)
- A tuple <a, b> representing the range [a, b].
The algorithm proceeds by iteratively constructing better approximations to finite presentations for G/K, where K is the intersection of kernels of all homomorphisms from G into Sn, and applying LowIndexSubgroups to the resulting finitely-presented group. The output information displayed for various values of the Print parameter about the number and existence of putative subgroups of index at most n refers to the current finite presentation only, may change as this presentation is further refined, and need not be reflected in the final answer.
Limit: RngIntElt Default: ∞Terminate after finding n conjugacy classes of subgroups satisfying the designated conditions.Print: RngIntElt Default: 0The Print parameter takes values from 0 to 3.
> G := GL (4, 5);
> L := LowIndexSubgroups (G, 4);
> #L;
3
> L[3];
MatrixGroup(4, GF(5))
Generators:
[4 0 0 4]
[1 0 0 0]
[0 4 0 0]
[0 0 4 0]
[4 0 0 3]
[3 0 0 0]
[0 4 0 0]
[0 0 4 0]
[4 0 0 1]
[4 0 0 0]
[0 4 0 0]
[0 0 4 0]
[4 0 0 2]
[2 0 0 0]
[0 4 0 0]
[0 0 4 0]
Representatives for the conjugacy classes of subgroups for the group G. The subgroups are returned as a sequence of records where the i-th record contains:
- (a)
- A representative subgroup H for the i-th conjugacy class (field name subgroup).
- (b)
- The order of the subgroup (field name order).
- (c)
- The number of subgroups in the class (field name length).
- (d)
- [Optional] A presentation for H (field name presentation).
Al: MonStgElt Default: "All"Al := "All": Construct all subgroups of G.Al := "Maximal": Only construct maximal subgroups of G. This option reduces the number of intersections with any elementary abelian layer that need be considered and eliminates the need to recursively apply the algorithm.
Al := "Normal": Only construct normal subgroups of G. This option does not use database lookup to find the normal subgroups of the radical quotient of G and also reduces the number of intersections with any layer that need be considered.
LayerSizes: SeqEnum Default: See belowLayerSizes := [ 2, 5, 3, 4, 7, 3, 11, 2, 17, 1] is equivalent to the default. When constructing an Elementary Abelian series for the group, attempt to split 2-layers of size gt 25, 3-layers of size gt 34, etc. The implied exponent for 13 is 2 and for all primes greater than 17 the exponent is 1.Series: SeqEnum Default: See belowUse the given elementary abelian series rather than constructing the default series. The first subgroup in the series must be the solvable radical of G. The subgroups must form a descending chain of normal subgroups of G, such that each quotient is elementary abelian. The last subgroup in the series must be either elementary abelian or trivial.Presentation: BoolElt Default: falsePresentation := true: Construct a presentation for each subgroup.OrderEqual: RngIntElt Default:OrderEqual := n: Only construct subgroups having order equal to n.OrderDividing: RngIntElt Default:OrderDividing := n: Only construct subgroups having order dividing n.OrderMultipleOf: RngIntElt Default:OrderMultipleOf := n: Only construct subgroups having order a multiple of n.IndexLimit: RngIntElt Default:IndexLimit := n: Only construct subgroups having index in G less than or equal to n.IsElementaryAbelian: BoolElt Default: falseIsElementaryAbelian := true: Only construct elementary abelian subgroups of G.IsCyclic: BoolElt Default: falseIsCyclic := true: Only construct cyclic subgroups of G.IsAbelian: BoolElt Default: falseIsAbelian := true: Only construct abelian subgroups of G.IsNilpotent: BoolElt Default: falseIsNilpotent := true: Only construct nilpotent subgroups of G.IsSolvable: BoolElt Default: falseIsSolvable := true: Only construct solvable subgroups of G.IsNotSolvable: BoolElt Default: falseIsNotSolvable := true: Only construct insolvable subgroups of G.IsPerfect: BoolElt Default: falseIsPerfect := true: Only construct perfect subgroups of G.The Algorithm: (See Cannon, Cox and Holt [CCH01]) This command proceeds by first constructing an elementary abelian series for G together with G's radical quotient Q as a permutation group. (Thus this function is limited to matrix groups over fields, where the group has a BSGS.) The required subgroups of Q are then found as for permutation groups. We first attempt to locate the quotient in a database of groups with trivial Fitting subgroup. This database contains all such groups of order up to 216 000, and all such which are perfect of order up to 1 000 000. If Q is found then either all its subgroups, or its maximal subgroups are read from the database. (In some cases only the maximal subgroups are stored.) If Q is not found then we attempt to find the maximal subgroups of Q using a method of Derek Holt. For this to succeed all simple factors of the socle of Q must be found in a second database which currently contains all simple groups of order less than 1.6 x 107, as well as M24, HS, J3, McL, Sz(32) and L6(2). There are also special routines to handle numerous other groups. These include: An for n ≤999, L2(q), L3(q), L4(q) and L5(q) for all q, U3(q) for q prime and q=8, 9, 16, 25, U4(q) for q=4, 5, 7, S4(q) for all odd q and even q≤16, Ld(2) for d ≤14, and the following groups: L6(3), L7(3), U6(2), S8(2), S10(2), O∓8(2), O∓10(2), S6(3), O7(3), O^ - 8(3), G2(4), G2(5), ()3D4(2), ()2F4(2)', Co2, Co3, He, Fi22.
If we have only maximal subgroups of Q, and more are required, we apply the algorithm recursively to the maximal subgroups to determine all subgroups of Q. This may take some time.
The subgroups of Q are then pulled back to G and extended to the whole group by stepwise extension through each layer of the elementary abelian series. For each layer this involves determining all possible intersections of a subgroup with this layer and all extensions with this intersection.
The limitations are that the simple factors of the socle of Q must be in the list above. Further, it may take some time to construct all subgroups from the maximal subgroups first found, and, if there is a large elementary abelian layer, there will be many possible intersections, which could also make the algorithm prohibitively slow.
There are numerous parameters for this function which allow the user to place restrictions on which subgroup classes are constructed. Using these restrictions may help overcome the problems noted above.
Construct the sequence of maximal subgroup classes of the matrix group G. This is equivalent to the command Subgroups(G: Al := "Maximal"). The same parameters as for Subgroups are available to limit the search.
Construct the sequence of maximal subgroup classes of G that contain the normal subgroup N of G. So this is equivalent to computing the maximal subgroups of G that contain N. Warning: Some parameters may have no effect.
This function isolates one step of the extension process used by the Subgroups family of functions. Q is a sequence of records such as returned by Subgroups(G). A and B are normal subgroups of G with A/B elementary abelian. The records in Q are interpreted as subgroups of G/A, which are lifted to all possible corresponding subgroups of G/B, subject to the parameters given.
Given a group G and subgroups H and K belonging to G, return the value true if H and K are conjugate in G. The function returns a second value in the event that the subgroups are conjugate: an element z which conjugates H into K.
Given H and K, both subgroups of the same general linear group G = GLn(q), return the value true if H and K are conjugate in G. The function returns a second value in the event that the subgroups are conjugate: an element z which conjugates H into K. The algorithm is described in Roney-Dougal [RD04].