Most of the group properties discussed in this section are usually hard to determine. Consequently, many of the intrinsics that test an fp-group G for a given property P only return a useful answer in those cases for which they are able to prove that G has property P. If they cannot do this then no conclusion can be drawn. The group property intrinsics that are available in Magma are listed below:-
Intrinsics:
IsPerfect (G) : Returns true if G is perfect and false if not. Note that this and the small cancellation intrinsic are the only ones in this list that can definitively decide whether or not the condition holds.
IsFinite (G) : Returns true if Magma is able to prove that G has finite order. Otherwise no conclusion can be drawn. If successful the order of G is also returned.
IsInfinite (G) : Returns true if Magma is able to prove that G is infinite. Otherwise no conclusion can be drawn.
IsLarge (G, l, u) : Attempts to show that G has a subgroup with homomorphic image the free group of rank 2. If successful, true is returned together with a subgroup witness to this property. If not successful, return false. The algorithm will consider subgroups of G with index h such that l ≤h ≤u.
SmallCancellationConditions (G) : Returns the small cancellation conditions T(i), C(j) and C'(k) that are satisfied by the presentation for G. Specifically it returns i, j, k', where T(i), C(j) and C'(k) are satisfied for all k > k'.
IsAutomaticGroup (G) : Returns true if Magma is able to prove that G is an automatic group. Otherwise no conclusion can be drawn. If successful the automatic structure for G is also returned.
IsHyperbolic (G) : Returns true if Magma is able to prove that G is a hyperbolic group. Otherwise no conclusion can be drawn. If successful a Dehn function for the group may be returned.
Notes:
Some of the above intrinsics are applied to the Coxeter group G = < x, y | x2, y3, (x * y)7, (x, y)9 > First a proof that the group is infinite is produced:-
> G<x,y> := FPGroup< x, y | x^2, y^3, (x * y)^7, (x, y)^9 >; > yes, proof := IsInfinite(G); > yes; true
So the group is infinite. The second return argument outlines a proof of this fact.
> proof; ! The following lines sketch a proof that the fp-group F is infinite:- ! F has the group PSL(2, 8) of order 504 as a quotient G (L2Quotient). ! Let k denote the rational field. ! Let M be the ext square of perm module of G as a kG-module. It has dim 36. ! Let N be the kF-module given by the induced action of F on M. ! The dimension of the first integral cohomology group of the kF-module N is 1. ! As H^1(F, N) is non-zero, the Holt-Plesken criterion shows that F is infinite.
Next the group is tested for being perfect.
> IsPerfect(G); true
So G is perfect. Next a subgroup of finite index in G is defined and tested for being perfect.
> H := sub< G | x, y * x * y * x * y * x * y^-1 >;
> Index(G, H);
9
> IsPerfect(H);
false
> AbelianQuotient(H);
Abelian Group isomorphic to Z/7
Defined on 1 generator
Relations:
7*$.1 = 0
So the subgroup H is not perfect and its derived subgroup has index 7 in H. Note that if H had infinite or very large index in G then the IsPerfect test would fail.
The final step is to determine the small cancellation conditions satisfied by G.
> a := SmallCancellationConditions(G); This presentation satisfies T( 3 ). This presentation satisfies C( 2 ). This presentation satisfies C'(k) for all k > 1/2.
The infinite dihedral group is defined and then its automatic structure is computed.
> G<a, b> := FPGroup< a, b | a^2, b^a = b^-1 >;
> G;
Finitely presented group G on 2 generators
Relations
a^2 = Id(G)
b^a = b^-1
> yes, A := IsAutomaticGroup(G);
> yes;
true
The second return value A is the group defined by the automatic structure of G. The growth function f for A is calculated. The fact that the group is infinite means that the growth function has to be a rational function rather than a polynomial. Consequently, it is necessary to compute the power series for the rational function f.
> R<x> := RationalFunctionField(Integers()); > f := GrowthFunction(A); > f; (-x^2 - 2*x - 1)/(x - 1)
This is a nice simple rational function. For a typical group the growth function has much larger degree numerator and denominator! The power series expansion of f is obtained by coercing f into the appropriate power series ring.
> PSR := PowerSeriesRing(Integers(): Precision := 15);
> s<t> := PSR!f;
> s;
1 + 3*t + 4*t^2 + 4*t^3 + 4*t^4 + 4*t^5 + 4*t^6 + 4*t^7 + 4*t^8 +
4*t^9 + 4*t^10 + 4*t^11 + 4*t^12 + 4*t^13 + 4*t^14 + O(t^15)
So there are three elements of G of length 1 and four elements
of length 2.