Finite FP-Groups

Finite fp-groups are often of interest and so a small number of intrinsics are provided for such groups.

Intrinsics:

IsFinite (G) : Returns true if G is proved to be finite, false if unknown. If G is found to be finite, its order is also returned.

Order (G) : Given a finite fp-group G return the order of G. If the order cannot be determined or is found to be infinite, then the value zero or Infinity, respectively, is returned.

PermutationGroup (G) : Given a finite fp-group G return a permutation group P that is isomorphic to G together with the isomorphism φ: G -> P.

CayleyGraph (G) : If G is an fp-group with generating set X return the Cayley graph CG of G with respect to the generating set X. The Cayley graph is returned as a Magma directed graph with its vertices and edges labelled. A parameter Labelled can be set to false to prevent the edges and vertices being labelled.

Notes:

Note that the intrinsics IsFinite, Order, PermutationGroup, and CayleyGraph may need to construct the coset table of G over the identity and hence are generally restricted to groups of order less than a million. The exact limit is determined by the amount of memory available.

Example GrpFPInt_OrderGroup (H77E22)

The intrinsic Order is invoked without any parameter settings to compute the order of the group G = < a, b | a8, b7, (ab)2, (ab - 1)3 > 
> G<a, b> := FPGroup<a,b | a^8, b^7, (a*b)^2, (a*b^-1)^3>;
> G;
Finitely presented group G on 2 generators
Relations
    a^8 = Id(G)
    b^7 = Id(G)
    (a * b)^2 = Id(G)
    (a * b^-1)^3 = Id(G)
> Order(G);
10752

Example GrpFPInt_CGraph (H77E23)

Now the Cayley graph of the group defined in the previous example is constructed. Next the greatest distance between any pair of vertices (the diameter) is found. 
> G<a, b> := FPGroup<a, b | a^8, b^7, (a*b)^2, (a*b^-1)^3>;
> CG := CayleyGraph(G);
> Diameter(CG);
22

Finally, something more sophisticated is demonstrated. First the shortest path g (geodesic) between vertex 1 and vertex 1000 is found. Then the labels on the edges of this path are multiplied together yielding the element of G corresponding to vertex 1000. 

> V := Vertices(CG);
> E := Edges(CG);
> g := Geodesic(V!1, V!1000);
> g;
[ 1, 2, 6, 16, 38, 33, 73, 82, 163, 309, 168, 318, 579, 999, 1642, 1000 ]
> w := &*[ Label(E![g[i], g[i+1]]) : i in [1.. #g-1] ];
> w;
a^5 * b * a^3 * b^4 * a * b
V2.28, 13 July 2023