Basic Operations#
Accessing Group Information#
The functions in this group provide access to basic information stored for a group \(G\).
- G . i: Grp, RngIntElt -> GrpElt#
The \(i\)-th defining generator for \(G\), if \(i>0\). If \(i<0\), then the inverse of the \(-i\)-th defining generator is returned. The generator
G.0is equivalent toIdentity(G).
- Generators(G): Grp -> { GrpFinElt }#
A set containing the defining generators for \(G\).
- NumberOfGenerators(G): Grp -> RngIntElt#
- Ngens(G): Grp -> RngIntElt#
The number of defining generators for \(G\).
- SmallestGeneratingSet(G: parameters): Grp -> SetIndx#
Print : RngIntElt Default: 0 QuickTries: RngIntElt Default: 250
A generating set of minimal cardinality for the finite group \(G\). The algorithm is from [Lucchini and Thakkar, 2024].
- Generic(G): Grp -> Grp#
Given a group \(G\) in the category
GrpPermorGrpMat, return the generic group containing \(G\), i.e., the largest group in which \(G\) is naturally embedded. The precise definition of generic group depends upon the category to which \(G\) belongs.
- Parent(g): GrpElt -> Grp#
The parent group \(G\) for the group element \(g\).
- Example: Generators (ex-a47283)#
The Suzuki simple group \(G={\operatorname{Sz}}(8)\) is constructed. Its generic group is \({\operatorname{GL}}(4, K)\), where \(K\) is the finite field with 8 elements. The field \(K\) is constructed first, so that its generator may be given the printname \(z\). Then the three generators of \(G\) are printed, in the standard order of indexing.
> K<z> := GF(2, 3); > G := SuzukiGroup(8); > Generic(G); GL(4, GF(2, 3)) > Ngens(G); 3 > for i in [1..3] do > print "generator", i, G.i; > print "order", Order(G.i), "\\r"; > end for; generator 1 [ 0 0 0 1] [ 0 0 1 0] [ 0 1 0 0] [ 1 0 0 0] order 2 generator 2 [z^2 0 0 0] [ 0 z^6 0 0] [ 0 0 z 0] [ 0 0 0 z^5] order 7 generator 3 [ 1 0 0 0] [z^2 1 0 0] [ 0 z 1 0] [z^5 z^3 z^2 1] order 4
- Orbit(G, M, x): Grp, Any, Any -> {Any}#
Given a finitely generated group \(G\) that acts on the parent structure of \(x\) through the map (or user defined function) \(M\), compute the orbit of \(x\) under \(G\). Thus, for every generator \(g\) of \(G\), \(M(g)\) must return a function that can be applied to \(x\) or any other element in the parent of \(x\).
If the orbit is infinite, this process will eventually run out of memory.
- OrbitClosure(G, M, S): Grp, Any, {Any} -> Any#
Given a finitely generated group \(G\) acting on the universe of \(S\) through the map or user defined function \(M\), compute the smallest subset \(T\) containing \(S\) that is \(G\)-invariant. Thus, for every generator \(g\) of \(G\), \(M(g)\) must return a function that can be applied to an arbitrary element in the universe of \(S\).
If the orbit closure is infinite, this process will eventually run out of memory.
Names of Finite Groups#
- GroupName(G): Grp -> MonStgElt#
TeX: BoolElt Default: false
Short name of a finite group \(G\), as an abstract group.
- Example: Grp Groupname (ex-7a2387)#
> [GroupName(G): G in SmallGroups(24)]; [ C3:C8, C24, SL(2,3), C3:Q8, C4*S3, D12, C2*C3:C4, C3:D4, C2*C12, C3*D4, C3*Q8, S4, C2*A4, C2^2*S3, C2^2*C6 ] > GroupName(AlternatingGroup(10): TeX:=true); A_{10}
Small groups (of order \(<512\), not divisible by \(128\)) have a unique name each, and Group(GroupName(G)) always returns a group isomorphic to \(G\). For larger groups, Magma attempts to recognize direct products, wreath products and split extensions, and uses chief series if that fails. With TeX:=true, the returned string is in LaTeX format. Note that for such larger groups the name returned is not canonical: isomorphic groups may be given different names.
Here is a list of notation used by GroupName. See also example below.
Basic groups
Cn Cyclic group of order n
Dn Dihedral group of order 2n
Sn Symmetric group on n letters
An Alternating group on n letters
Operators, high to low precedence
^ power, e.g C2^2 is the non-cyclic group of order 4
wr wreath product, e.g. C2wrC2=C2^2:C2=D4
: semidirect product, i.e. a split extension
. (generally) non-split extension
* direct product
Other standard groups
Fq Frobenius group of order q(q-1)
Hep Heisenberg group of order p^3
Qn Generalized quaternion group, n=2^k
SDn Semi-dihedral group C2^(k-1):C2 (n=2^k) with C2
acting as 2^(k-2)-1
ODn Other-dihedral group C2^(k-1):C2 (n=2^k) with C2
acting as 2^(k-2)+1
Simple, almost-simple and linear groups
Mn Matthieu group (n in {11,12,21,22,23,24})
GL(n,q) General linear group; also SL,AGL,ASL,AGammaL,ASigmaL,PGL,
PSL (=L),PGammaL,PSigmaL,SU,PSU,PGammaU,PSigmaU,O (=GO),SO,
PSO,PGO,PGO+,PGO-,POmega,POmega+,POmega-,Sp,PSp,PSigmaSp
B(n,q) Simple group of Lie type, also C,D,E,F,G,2A,2B,2D,2E,2F,2G,3D
J1 Sporadic simple group; also Mn (see above),J2,J3,J4,HS,McL,Suz,
Co1,Co2,Co3,HE,Fi22,Fi23,Fi24,Ly,Ru,ON,TH,HN,BM,M
- Group(s): MonStgElt -> Grp#
A finite group from its name. See
GroupNameand the example below.
- Example: Grp Group (ex-06353f)#
> G0:=Group("C10^2*C3"); // cyclic and abelian > G1:=Group("D5"); // dihedral Dn of order 2n > G2:=Group("A5"); // alternating > G3:=Group("S5"); // symmetric > > G4:=Group("SL(2,3)"); // linear: GL, SL, AGL, ASL, AGammaL, ASigmaL, PGL, > G5:=Group("SL(2,F3)"); // PSL (=L), PGammaL, PSigmaL, SU, PSU, PGammaU, > G6:=Group("SL_2(3)"); // PSigmaU, O (=GO), SO, PSO, PGO, PGO+, PGO-, > G7:=Group("SL2(3)"); // POmega, POmega+, POmega-, Sp, PSp, PSigmaSp > > G8:=Group("S3*GL(4,2)"); // Products > G9:=Group("C41:C40"); // Split extensions that are not direct products, > // [usually with largest action of the quotient group] > G10:=Group("A5wrC2"); // Wreath products > > G11:=Group("C2^3.C4"); // unique names returned by GroupName > // when |G|<512, not multiple of 128 > G12:=Group("A5*A_5*A_{5}*Alt(5)"); // name variations > G13:=Group("D10:C8.C2*C3"); // operator order ^ > wr > : > . > * > // (so read left to right in this example) > > G14:=Group("<12,1>"); // Small group database (C3:C4) > G14:=Group("g12n1"); // same group > G15:=Group("T<12,48>"); // Transitive group database (C2^2*S4) > G15:=Group("t12n48"); // same group > // Simple groups: Lie Type A,B,C,D,E,F,G, returned > G16:=Group("C(4,2)"); // as matrix groups via standard representation Warning: Projective representation > G17:=Group("Sz(32)"); // Simple groups: Suzuki > G18:=Group("J1*Co3*M11"); // Simple groups: sporadic > G19:=Group("PGL(4,3)`2"); // Names from the almost simple group database > > G20:=Group("He11"); // Heisenberg > G21:=Group("F13"); // Frobenius group Fn of order n(n-1) > G22:=Group("Q8"); // Quasi-cyclic groups of normal 2-rank one: > G23:=Group("SD16"); // Dihedral, (generalized) quaternion, > G24:=Group("OD16"); // semi-dihedral, the `other-dihedral' one. > > [GroupName(eval "G"*Sprint(n)): n in [1..24]]; // back to names [ D5, A5, S5, SL(2,3), SL(2,3), SL(2,3), SL(2,3), S3*A8, F41, A5wrC2, C2^3.C4, A5^4, C3*D10:C8.C2, C3:C4, C2^2*S4, C(4,2), 2B(2,32), J1*Co3*M11, PSL(4,3).C2^2, He11, F13, Q8, SD16, OD16 ]