Construct the centre of the group G.
Construct the hypercentre of the group G (the stationary term of the upper central series).
The derived length of G. If G is non-soluble, the function returns the number of terms in the series terminating with the soluble residual.
The derived series of the group G. The series is returned as a sequence of subgroups.
The derived subgroup of the group G.
The Fitting subgroup of the group G.
Given a group G that is a p-group, return the Frattini subgroup.
Given a p-group G, return the Jennings series for G. The series is returned as a sequence of subgroups.
The lower central series of G. The series is returned as a sequence of subgroups.
A maximal normal subgroup of G.
The minimal normal subgroups of G. If G is primitive, this function is quite efficient. However, if G is intransitive or imprimitive, the minimal normal subgroups are found by computing the conjugacy classes of G and looking at the normal subgroups generated by each class. In this situation, the function is restricted to groups of moderate order.
The nilpotency class of the group G. If the group is not nilpotent, the value -1 is returned.
The normal closure of the subgroup H of group G.
The normal subgroup lattice of G. The subgroups are found by first determining the minimal normals using the conjugacy classes of G and then extending these layer by layer until G is reached. The algorithm is thus only practical in the case of small to medium sized groups.
The normal subgroups of G.
Given a soluble group G, and a prime p dividing |G|, return the lower p-central series for G. The series is returned as a sequence of subgroups.
The solvable residual of the group G.
Given a group G and a subnormal subgroup H of G, return a sequence of subgroups commencing with G and terminating with H, such that each subgroup is normal in the previous one. If H is not subnormal in G, the empty sequence is returned.
The upper central series of G. The series is returned as a sequence of subgroups commencing with the trivial subgroup. Since the algorithm used requires the conjugacy classes of G, this function is much more restricted in its range of application than DerivedSeries and LowerCentralSeries.
> G := WreathProduct(Sym(4), DihedralGroup(4));
> print G;
Permutation group G acting on a set of cardinality 16
(1, 5, 9, 13)(2, 6, 10, 14)(3, 7, 11, 15)(4, 8, 12, 16)
(1, 13)(2, 14)(3, 15)(4, 16)(5, 9)(6, 10)(7, 11)(8, 12)
(1, 2, 3, 4)
(1, 2)
> print DerivedSeries(G);
[
Permutation group G acting on a set of cardinality 16
Order = 2654208 = 2^15 * 3^4
(1, 5, 9, 13)(2, 6, 10, 14)(3, 7, 11, 15)(4, 8, 12, 16)
(1, 13)(2, 14)(3, 15)(4, 16)(5, 9)(6, 10)(7, 11)(8, 12)
(1, 2, 3, 4)
(1, 2),
Permutation group acting on a set of cardinality 16
Order = 331776 = 2^12 * 3^4
(1, 9)(2, 10)(3, 11)(4, 12)(5, 13)(6, 14)(7, 15)(8, 16)
(1, 12)(2, 9)(3, 10)(4, 11)(5, 13)(6, 14)(7, 15)(8, 16)
(2, 4, 3)(9, 11, 12)
(6, 7, 8)(14, 16, 15)
(5, 6, 7, 8)(9, 12, 11, 10)
(6, 8, 7)
(9, 12, 11, 10)(13, 14, 15, 16)
(10, 11, 12)
(2, 3)(9, 10, 11, 12)
(3, 4)(7, 8)
(2, 3, 4)(6, 8, 7)
(3, 4)(6, 7, 8)(15, 16)
(3, 4)(11, 12),
Permutation group acting on a set of cardinality 16
Order = 41472 = 2^9 * 3^4
(5, 7)(6, 8)(13, 15)(14, 16)
(1, 3)(2, 4)(9, 11)(10, 12)
(1, 4)(2, 3)(9, 11)(10, 12)
(2, 3, 4)(10, 12, 11)
(3, 4)(6, 7)(10, 12)(15, 16)
(6, 8, 7)(14, 15, 16)
(6, 7, 8)(10, 12, 11)
(2, 4)(7, 8)(9, 12, 11, 10)(14, 15)
(6, 7, 8)(13, 16, 15)
(10, 11, 12),
Permutation group acting on a set of cardinality 16
Order = 20736 = 2^8 * 3^4
(1, 2)(3, 4)(9, 12)(10, 11)
(5, 8)(6, 7)(13, 14)(15, 16)
(9, 10)(11, 12)
(5, 6)(7, 8)
(1, 4)(2, 3)(9, 10)(11, 12)
(13, 16)(14, 15)
(9, 11)(10, 12)
(13, 14)(15, 16)
(6, 7, 8)(14, 16, 15)
(10, 12, 11)
(2, 4, 3)(10, 12, 11)
(6, 7, 8)(10, 12, 11),
Permutation group acting on a set of cardinality 16
Order = 256 = 2^8
(13, 14)(15, 16)
(5, 6)(7, 8)
(9, 10)(11, 12)
(5, 7)(6, 8)(13, 14)(15, 16)
(13, 15)(14, 16)
(1, 3)(2, 4)(9, 10)(11, 12)
(1, 4)(2, 3)(9, 10)(11, 12)
(9, 12)(10, 11),
Permutation group acting on a set of cardinality 16
Order = 1
Id($)
]
> print DerivedLength(G);
5
> print [ FactoredOrder(H) : H in LowerCentralSeries(G) ];
[
[ <2, 15>, <3, 4> ],
[ <2, 12>, <3, 4> ],
[ <2, 10>, <3, 4> ],
[ <2, 9>, <3, 4> ],
[ <2, 8>, <3, 4> ]
]
> print NilpotencyClass(G);
-1
> print Centre(G);
Permutation group acting on a set of cardinality 16
Order = 1
Id($)
> print pCentralSeries(G, 2);
[
Permutation group G acting on a set of cardinality 16
Order = 2654208 = 2^15 * 3^4
(1, 5, 9, 13)(2, 6, 10, 14)(3, 7, 11, 15)(4, 8, 12, 16)
(1, 13)(2, 14)(3, 15)(4, 16)(5, 9)(6, 10)(7, 11)(8, 12)
(1, 2, 3, 4)
(1, 2),
Permutation group acting on a set of cardinality 16
Order = 331776 = 2^12 * 3^4
(1, 9)(2, 10)(3, 11)(4, 12)(5, 13)(6, 14)(7, 15)(8, 16)
(1, 2, 3, 4)(5, 8, 7, 6)
(1, 2)(5, 6)
(1, 2, 3, 4)(13, 16, 15, 14),
Permutation group acting on a set of cardinality 16
Order = 82944 = 2^10 * 3^4
(1, 2, 3, 4)(5, 7)(6, 8)(9, 10, 11, 12)
(1, 3)(2, 4)(5, 8, 7, 6)(13, 16, 15, 14)
(1, 2)(9, 10)
(1, 2)(5, 6)(9, 10)(13, 14)
(1, 2, 3)
(5, 6, 7),
Permutation group acting on a set of cardinality 16
Order = 41472 = 2^9 * 3^4
(1, 2, 3, 4)(5, 6, 7, 8)(9, 12, 11, 10)(13, 16, 15, 14)
(1, 2, 3)(5, 7, 6)
(1, 4, 3, 2)(5, 6, 7, 8)(9, 10, 11, 12)(13, 16, 15, 14)
(5, 6, 7)(9, 11, 10)
(1, 2, 3)(13, 15, 14)
(1, 2)(5, 6)(9, 10)(13, 14),
Permutation group acting on a set of cardinality 16
Order = 20736 = 2^8 * 3^4
(1, 3)(2, 4)(9, 11)(10, 12)
(1, 2, 3)(5, 6, 7)(9, 10, 11)
(5, 6, 7)(9, 10, 11)(13, 14, 15)
(1, 3, 2)(5, 7, 6)(13, 15, 14)
(1, 2, 4)
(5, 7)(6, 8)(13, 15)(14, 16)
]
The elementary abelian regular normal subgroup (EARNS) of the primitive group G. If G does not have a proper EARNS, then the trivial subgroup is returned.
The socle of the group G.
A simple factor of the socle of the primitive group G.
The set of simple factors of the socle of the primitive group G.
A chain of subgroups S_1, S_1 x S_2, ..., S_1 x ... x S_r, where S_1, ..., S_r are the simple factors of the socle of the primitive group G.
> G := TensorProduct(Sym(5), Sym(3));
> print FactoredOrder(G);
[ <2, 10>, <3, 4>, <5, 3> ]
> E := EARNS(G);
> print E;
Permutation group E acting on a set of cardinality 125
Order = 1
> print DerivedSeries(G);
[
Permutation group G acting on a set of cardinality 125
Order = 10368000 = 2^10 * 3^4 * 5^3
Permutation group acting on a set of cardinality 125
Order = 2592000 = 2^8 * 3^4 * 5^3,
Permutation group acting on a set of cardinality 125
Order = 864000 = 2^8 * 3^3 * 5^3,
Permutation group S acting on a set of cardinality 125
Order = 216000 = 2^6 * 3^3 * 5^3
]
> S := Socle(G);
> print S;
Permutation group S acting on a set of cardinality 125
Order = 216000 = 2^6 * 3^3 * 5^3
> Q := SocleFactors(G);
> print Q;
[
Permutation group acting on a set of cardinality 125
Order = 60 = 2^2 * 3 * 5,
Permutation group acting on a set of cardinality 125
Order = 60 = 2^2 * 3 * 5,
Permutation group acting on a set of cardinality 125
Order = 60 = 2^2 * 3 * 5
]
> R := SocleSeries(G);
> print R;
[
Permutation group acting on a set of cardinality 125
Order = 60 = 2^2 * 3 * 5,
Permutation group acting on a set of cardinality 125
Order = 3600 = 2^4 * 3^2 * 5^2,
Permutation group acting on a set of cardinality 125
Order = 216000 = 2^6 * 3^3 * 5^3
]
centerline{hfill
vbox to 15truecm{l offinterlineskip
hrule halign{&vrule#& hfil strut#hfil &vrule#& strut#hfil cr
&f&&hfil Family name&cr
noalign{hrule}
& 1 && A(d, q) &cr
& 2 && B(d, q) &cr
& 3 && C(d, q) &cr
& 4 && D(d, q) &cr
& 5 && G(2, q) &cr
& 6 && F(4, q) &cr
& 7 && E(6, q) &cr
& 8 && E(7, q) &cr
& 9 && E(8, q) &cr
& 10 && 2A(d, q) &cr
& 11 && 2B(2, q) &cr
& 12 && 2D(d, q) &cr
& 13 && 3D(4, q) &cr
& 14 && 2G(2, q) &cr
& 15 && 2F(4, q) &cr
& 16 && 2E(6, q) &cr
& 17 && Alternating(d) &cr
& 18 && Sporadic group --- see Table 2. & cr
& 19 && Cyclic(q) &cr
}hrulel}hfill
vbox to 15truecm{l offinterlineskip
hrule halign{&vrule#& hfil strut#hfil &vrule#& strut#hfil cr
&d&&hfil Group name&cr
noalign{hrule}
& 1 && M_(11) &cr
& 2 && M_(12) &cr
& 3 && M_(22) &cr
& 4 && M_(23) &cr
& 5 && M_(24) &cr
& 6 && J_1 &cr
& 7 && HS &cr
& 8 && J_2 &cr
& 9 && MCL &cr
& 10 && SUZ &cr
& 11 && J_3 &cr
& 12 && CO_1 &cr
& 13 && CO_2 &cr
& 14 && CO_3 &cr
& 15 && HE &cr
& 16 && M(22) &cr
& 17 && M(23) &cr
& 18 && M(24) &cr
& 19 && LY &cr
& 20 && RU &cr
& 21 && ON &cr
& 22 && TH &cr
& 23 && HA &cr
& 24 && BM &cr
& 25 && M &cr
& 26 && J_4 &cr
& 27 && TITS &cr
}hrulel}hfill}
centerline hfill{Table 1: Family numbers and names hfill{Table 2: Sporadic groups}hfill}
AbelianInvariants(G) : GrpPerm -> [ RngIntElt ]
Given an abelian group G, return a sequence Q containing the types of each p-primary component of G.
Given a permutation group G, return a sequence S of tuples that
represent the composition factors of G, ordered according to some
composition series of G. Each tuple is a triple
of integers f, d, q that defines the isomorphism type of
the corresponding composition factor. A triple < f, d, q >
describes a simple group as follows. The integer f defines
the family to which the group belongs, and d and q are the
parameters of the family. The length of the sequence S is the
number of composition factors of G.
The families are:
f family name
-------------------------
1 A(d, q)
2 B(d, q)
3 C(d, q)
4 D(d, q)
5 G(2, q)
6 F(4, q)
7 E(6, q)
8 E(7, q)
9 E(8, q)
10 2A(d, q)
11 2B(2, q)
12 2D(d, q)
13 3D(4, q)
14 2G(2, q)
15 2F(4, q)
16 2E(6, q)
17 Alternating(d)
18 Sporadic group --- see next list
19 Cyclic(q)
For f=18, the sporadic groups are:
d group name
-------------------------
1 M_11
2 M_12
3 M_22
4 M_23
5 M_24
6 J_1
7 HS
8 J_2
9 MCL
10 SUZ
11 J_3
12 CO_1
13 CO_2
14 CO_3
15 HE
16 M(22)
17 M(23)
18 M(24)
19 LY
20 RU
21 ON
22 TH
23 HA
24 BM
25 M
26 J_4
27 TITS
Given a simple group G, determine the isomorphism type of G. The type is returned in the form of a triple of three integers f, d and q, where the interpretation of these integers is that given in the description of the function CompositionFactors.
> G := sub<Sym(48) |
> (1,3,8,6)(2,5,7,4)(9,48,15,12)(10,47,16,13)(11,46,17,14),
> (6,15,35,26)(7,22,34,19)(8,30,33,11)(12,14,29,27)(13,21,28,20),
> (1,12,33,41)(4,20,36,44)(6,27,38,46)(9,11,26,24)(10,19,25,18),
> (1,24,40,17)(2,18,39,23)(3,9,38,32)(41,43,48,46)(42,45,47,44),
> (3,43,35,14)(5,45,37,21)(8,48,40,29)(15,17,32,30)(16,23,31,22),
> (24,27,30,43)(25,28,31,42)(26,29,32,41)(33,35,40,38)(34,37,39,36)
> >;
> print FactoredOrder(G);
[ <2, 27>, <3, 14>, <5, 3>, <7, 2>, <11, 1> ]
> print CompositionFactors(G);
G
| Cyclic(2)
*
| Alternating(12)
*
| Cyclic(2)
*
| Cyclic(2)
*
| Cyclic(2)
*
| Cyclic(2)
*
| Cyclic(2)
*
| Cyclic(2)
*
| Cyclic(2)
*
| Cyclic(2)
*
| Cyclic(2)
*
| Cyclic(2)
*
| Cyclic(2)
*
| Alternating(8)
*
| Cyclic(3)
*
| Cyclic(3)
*
| Cyclic(3)
*
| Cyclic(3)
*
| Cyclic(3)
*
| Cyclic(3)
*
| Cyclic(3)
1