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 \(\phi: G \rightarrow P\).

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: Order Group (ex-fb0cc3)

The intrinsic Order is invoked without any parameter settings to compute the order of the group

\[G =\,\, \langle a, b\ |\ a^8, b^7, (ab)^2, (ab^{-1})^3 \rangle\]

> 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

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Example: C Graph (ex-19bd36)

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

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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

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