Magma includes the Small Groups Library prepared by Besche, Eick and O'Brien. For a description of the algorithms used to generate these groups, details on the data structures used and applications we refer to [BE99a], [BEO01], [BE99b], [O'B90], [BE01], [O'B91], [MNVL04], [DEP22] and the references therein.
The Small Groups Library contains the following groups.
The Magma version of this library uses the same internal data format as the implementation available in GAP. In particular, the numbering of the groups of a given order in both packages is the same.
For a different mechanism for accessing the p-groups in this collection, see the The p-groups of Order Dividing p7 section, specifically the functions SearchPGroups and CountPGroups. These functions also access the groups of order p7 (contributed by O'Brien and Vaughan-Lee).
The groups of order 38 (also contributed by O'Brien and Vaughan-Lee) form an optional database that must be downloaded separately.
Many of the functions in this section have an optional parameter Search. It can be used to limit the small group search to soluble (Search := "Soluble") or insoluble (Search := "Insoluble") groups. The default is Search := "All", which allows all groups to be considered.
When a group is extracted from the database, it is returned as a GrpPC if it is soluble, or as a GrpPerm if it is insoluble.
When using the small groups database for an extended search, it is advisable to open the database using the function SmallGroupDatabase, which opens the database and returns a reference to it. This reference can then be passed as first argument to most of the functions described below, and will save that function from opening and closing the database for itself. Doing so will reduce the number of file operations when a lot of use is made of the database. When the database is no longer needed, it can be closed using the delete statement.
Open the small groups database (for extended search) and return a reference to it. This reference may be passed to other functions so that they do fewer file operations.
Close the small groups database D and free the resources associated with its use.
The limiting order up to which all groups (except those of order 1024) are stored in the database of small groups, that is, currently 2000.
Return true if the groups of order o are contained in the database and false otherwise. This function can be used to check whether o is a legitimate argument for other functions described in this section, avoiding runtime errors in user written loops or functions.
Given a positive integer o, return the number of groups of order o in the database. If the groups of order o are not contained in the database, 0 is returned. This function can be used to check whether a pair o, n defines a group contained in the small groups database, that is, whether it is a legitimate argument for other functions described in this section, avoiding runtime errors in user written loops or functions.
Given a positive integer o, such that the groups of order o are contained in the small groups library, and a positive integer n, return the n-th group of order o in the database. If the groups of order o are not contained in the database or if n exceeds the number of groups of order o in the database, an error is reported. The function NumberOfSmallGroups can be used to check whether the arguments are valid.
Search: MonStgElt Default: "All"
Given a positive integer o, such that the groups of order o are contained in the small groups library, return the first group of order o in the database meeting the search criterion set by the parameter Search. If the groups of order o are not contained in the database, an error is reported. The function IsInSmallGroupDatabase can be used to check whether o is a valid argument for this function.
Search: MonStgElt Default: "All"
Given a positive integer o such that the groups of order o are contained in the small groups library and a predicate f (as a function or intrinsic), return the first group of order o in the database meeting the search criterion set by the parameter Search, which satisfies f.
Return true iff SmallGroup(o, n) is soluble. This function does not load the group. If the group specified by the arguments does not exist in the database, an error is reported. The function NumberOfSmallGroups can be used to check whether the arguments are valid.
Return true iff SmallGroup(o, n) is insoluble. This function does not load the group. If the group specified by the arguments does not exist in the database, an error is reported. The function NumberOfSmallGroups can be used to check whether the arguments are valid.
Search: MonStgElt Default: "All"
Given a sequence S of orders or a single order o contained in the database and a predicate f (as a function or intrinsic), return the first group with order in S or equal to o, respectively, which meets the search criterion set by the parameter Search and satisfies f.
Search: MonStgElt Default: "All"
Warning: BoolElt Default: true
Given an order o contained in the database, return a list of all groups of order o, meeting the search criterion set by the parameter Search. Some orders will produce a very large sequence of groups -- in such cases a warning will be printed unless the user specifies Warning := false.
Search: MonStgElt Default: "All"
Warning: BoolElt Default: true
Given a sequence S of orders contained in the database, return a list of all groups with order in S, meeting the search criterion set by the parameter Search. The resulting sequence may be very long -- in such cases a warning will be printed unless the user specifies Warning := false.
Search: MonStgElt Default: "All"
Given an order o contained in the database and a predicate (function or intrinsic) f, return a list containing all groups G of order o, meeting the search criterion set by the parameter Search and satisfying f(G) eq true.
Search: MonStgElt Default: "All"
Given a sequence S of orders contained in the database and a predicate (function or intrinsic) f, return a list containing all groups G with order in S, meeting the search criterion set by the parameter Search and satisfying f(G) eq true.
> list := SmallGroups(27, func<x|not IsAbelian(x)> ); > list; [* GrpPC of order 27 = 3^3 PC-Relations: $.2^$.1 = $.2 * $.3, GrpPC of order 27 = 3^3 PC-Relations: $.1^3 = $.3, $.2^$.1 = $.2 * $.3 *]
> G := SmallGroup([1..100], func<x|DerivedLength(x) gt 2>); > G; GrpPC of order 24 = 2^3 * 3 PC-Relations: G.1^3 = Id(G), G.2^2 = G.4, G.3^2 = G.4, G.4^2 = Id(G), G.2^G.1 = G.3, G.3^G.1 = G.2 * G.3, G.3^G.2 = G.3 * G.4
> list := SmallGroups(240:Search:="Insoluble"); > #list; 8 > list[7]; Permutation group acting on a set of cardinality 7 (1, 2, 3, 4) (1, 5, 2, 4, 3)(6, 7)
> IsInSmallGroupDatabase(2432); true... and determine the number of groups of order 2432.
> NumberOfSmallGroups(2432); 19324
> f := function (G) > Z := Centre (G); > return IsCyclic (Z) and #Z eq 7^2; > end function; > P := SmallGroups(7^6, f); > #P; 30 > NumberOfSmallGroups(7^6); 860
A small group process makes it possible to iterate over all groups of specified orders satisfying a given predicate, without having to create and store all such groups in a list.
A small group process is created via the function SmallGroupProcess (in various forms). The standard process functions IsEmpty, Current, CurrentLabel and Advance can then be applied to the process.
The functions used to create a small group process all have a parameter Search attached to them. This can be used to limit the small group search to soluble (Search := "Soluble") or insoluble (Search := "Insoluble") groups. The default is Search := "All", which allows all groups to be considered.
The Process functions described below do not have a variant with the database as first argument, as each process opens the database for an extended search automatically.
Search: MonStgElt Default: "All"
Given an order o contained in the small groups database, return a small group process which will iterate though all groups of order o meeting the search criterion set by the parameter Search.
Search: MonStgElt Default: "All"
Given a sequence S of orders contained in the small groups database, return a small group process which will iterate though all groups with order in the sequence S meeting the search criterion set by the parameter Search.
Search: MonStgElt Default: "All"
Given an order o contained in the small groups database and a predicate f (as function or intrinsic), return a small group process which will iterate though all groups of order o, which meet the search criterion set by the parameter Search and satisfy the predicate f.
Search: MonStgElt Default: "All"
Given a sequence S of orders contained in the small groups database and a predicate f (as function or intrinsic), return a small group process which will iterate though all groups with order in the sequence S, which meet the search criterion set by the parameter Search and satisfy the predicate f.
Returns true if the process p has passed its last group.
Return the current group of the process p.
Return the label of the current group of the process p. That is, return o and n such that the current group is SmallGroup(o, n).
Move the process p to its next group.
> P := SmallGroupProcess(128);
> count := {* *};
> repeat
> G := Current(P);
> Include(~count, NilpotencyClass(G));
> Advance(~P);
> until IsEmpty(P);
> count;
{* 1^^15, 2^^947, 3^^1137, 4^^197, 5^^29, 6^^3 *}
Intrinsics are provided that allow the user to find the database group that is isomorphic to a given group. Given a group G such that the database includes a group isomorphic to G and assuming that identification of groups of order |G| is supported, the identification functions return a pair < o, n > so that SmallGroup(o, n) is isomorphic to G.
Identifying a finitely presented group G involves constructing a permutation representation G which may fail. The identification process tries to construct a permutation representation of G by enumerating the cosets of the trivial subgroup in G. Assuming that a group isomorphic to G is contained in the database, the resulting coset table will be fairly small. Hence for performance reasons, a coset limit of 100.N is imposed, where N is the largest order of groups in the database. If the order of G is known to be less or equal to N, the implicitly invoked coset enumeration can be given a more appropriate value for the maximum number of cosets that can be defined. This is done by means of the intrinsic SetGlobalTCParameters.
If the coset enumeration for G fails with the coset limit set to 100.N, this can be seen as a reasonable indication that G is probably too large to be includes in the database of small groups.
To deal with cases where the coset enumeration fails when G is known or suspected to be included in the database, the user should attempt to compute the order of G using the intrinsic Order prior to the actual group identification. If this shows G is small enough to be included in the database, the intrinsic SetGlobalTCParameters can be used to set a larger limit on maximum number of cosets that can be defined by the Todd-Coxeter procedure in a subsequent call to IdentifyGroup. This is illustrated in the following example.
Locate the pair of integers < o, n > so that SmallGroup(o, n) is isomorphic to G. If there is no such group in the database or if identification of groups of order |G| is not supported, then an error will result. The function CanIdentifyGroup can be used to test whether groups of a certain order can be identified; this may be useful for avoiding runtime errors in user written loops or functions.
Return true if identification of groups of order o in the database is supported. This function can be used to check whether a group is a legitimate argument for the functions IdentifyGroup described above, avoiding runtime errors in user written loops or functions.
> G := DihedralGroup(10);
> G;
Permutation group G acting on a set of cardinality 10
Order = 20 = 2^2 * 5
(1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
(1, 10)(2, 9)(3, 8)(4, 7)(5, 6)
> IdentifyGroup(G);
<20, 4>
> H := SmallGroup(20, 4);
> H;
GrpPC : H of order 20 = 2^2 * 5
PC-Relations:
H.1^2 = Id(H),
H.2^2 = Id(H),
H.3^5 = Id(H),
H.3^H.1 = H.3^4
Consider the group defined by the presentation < a, b | (ba - 1)3, (ba - 1b)2, a12b2a7b2ab2 >.
> G := Group<a,b | (b*a^-1)^3, (b*a^-1*b)^2, > a^12*b^2*a^7*b^2*a*b^2 >;We suspect (for some reason) that G is a small group and want to identify its isomorphism type in the database of small groups.
> IdentifyGroup(G);
IdentifyGroup(
G: GrpFP: G
)
In file "/home/magma/package/Group/Grp/smallgps2.m", line 220,
column 25:
>> res := IdentifyGroup(db, G);
^
Runtime error in 'IdentifyGroup': Coset enumeration failed; group
may be too large (see handbook entry for details)
The group couldn't be identified, because it was not possible to construct a
permutation representation with the default value of the coset limit.
Since we still believe the group is small, we try to prove this using
the function Order.
> Order(G : Print := true); INDEX = 6 (a=6 r=66709 h=999999 n=999999; l=1247 c=1.56; m=969169 t=999998) 6The output line beginning with "INDEX" gives information about the coset enumeration: in particular 999,998 cosets had to be designed in order to construct the coset table. We were correct; the group has order 6. However, we can see from the verbose output of the Order intrinsic that this wasn't easy to do: almost one million cosets were used in the coset enumeration. Since the order of G is now known to Magma, a subsequent call to IdentifyGroup will use 100*6 as the coset limit. We tell the system to be prepared to work a little harder using the intrinsic SetGlobalTCParameters and again try to identify G:-
> SetGlobalTCParameters( : Strategy := "Hard"); > > IdentifyGroup(G); <6, 1>So G has now been identified!
The following functions provide access to data used internally by the Small Groups Library for representing groups. They are included just for completeness and are intended to be used by experts only. In particular, we do not give a detailed explanation of the (complicated) data format in this manual.
Returns the data from which the group number n of order o in D is constructed. The format and the meaning of the items in the returned list depends on the group indicated by the pair (o, n).
Given a finite solvable group G in the category GrpPC, return two integers c and o encoding the power conjugate presentation of G.The second return value is the order of G. The first return value is an integer specifying the power conjugate relations in the presentation of G.
Given two integers c and o encoding a power conjugate presentation G, return G as a group in the category GrpPC.The second argument is the order of G. The first return value is an integer specifying the power conjugate relations in the presentation G.
> D := SmallGroupDatabase(); > G := SmallGroup(D,1053,51); > Category(G); GrpPC > SmallGroupEncoding(G); 100286712487397165939678173 1053
> Data(D,525,2); [* code, 666501 *]We can create the corresponding group by decoding this information.
> G := SmallGroupDecoding(666501, 525);
> G;
GrpPC : G of order 525 = 3 * 5^2 * 7
PC-Relations:
G.1^3 = Id(G),
G.2^5 = G.4,
G.3^7 = Id(G),
G.4^5 = Id(G)
This gives the same presentation as accessing the database in the "usual"
way.
> SmallGroup(D,525,2);
GrpPC of order 525 = 3 * 5^2 * 7
PC-Relations:
$.1^3 = Id($),
$.2^5 = $.4,
$.3^7 = Id($),
$.4^5 = Id($)
The following functions provide access to functions used internally by the Small Groups Library for constructing groups of order divisible by four primes. These functions are based on the paper [DEP22] by Dietrich, Eick, and Pan. This article gives constructions for all groups of orders p2q, p3q, p2q2, and p2qr, where p, q, r are distinct primes. These constructions have been implemented in Magma and are available as follows. This includes a function to identify such a group in the lists. In all the functions here, the primes p, q, r must be < 230.
The functions here are used by the Small Groups Library when the group order is as above, except when the order is less than 2000, or has the form p2q, or the form p3q with p≤7 (which were in the small groups library before these functions were written). In these exceptional cases the numbering of the groups within each order usually differs between the Small Groups Library and the functions here.
The groups are constructed as pc-groups, except in one case: order 60, group number 13 is the only insoluble group within these orders (it is A5), and this group is returned as a permutation group.
Returns the ith group in the list of groups of order o given by [DEP22]. The order o must have the form p2q, p3q, p2q2, or p2qr, where p, q, r are distinct primes.
Returns the list of groups of order o given by [DEP22]. The order o must have the form p2q, p3q, p2q2, or p2qr, where p, q, r are distinct primes.
Returns the length of the list of groups of order o given by [DEP22]. The order o must have the form p2q, p3q, p2q2, or p2qr, where p, q, r are distinct primes.
Returns the number of the group isomorphic to G in the the list of groups of order o=|G| given by [DEP22]. The order o must have the form p2q, p3q, p2q2, or p2qr, where p, q, r are distinct primes. Note that this number may not agree with the number returned by texttt{IdentifyGroup} (see above).