Operations on the Set of Group Elements#

Random(G, n): GrpRWS, RngIntElt -> GrpRWSElt#: A random word of length at most \(n\) in the generators of \(G\).

Random(G): GrpRWS -> GrpRWSElt#: A random word (of length at most the order of \(G\)) in the generators of \(G\).

Representative(G): GrpRWS -> GrpRWSElt#
Rep(G): GrpRWS -> GrpRWSElt#: An element chosen from \(G\).

Set(G, a, b): GrpRWS, RngIntElt, RngIntElt -> SetEnum#

Search: MonStgElt                    Default: "DFS"

Create the set of reduced words, \(w\), in \(G\) with \(a \le\) length\((w) \le b\). If Search is set to "DFS" (depth-first search) then words are enumerated in lexicographical order. If Search is set to "BFS" (breadth-first-search) then words are enumerated in lexicographical order for each individual length (i.e. in short-lex order). Depth-first-search is marginally quicker. Since the result is a set the words may not appear in the resultant set in the search order specified (although internally they will be enumerated in this order).

Set(G): GrpRWS -> SetEnum#

Search: MonStgElt                    Default: "DFS"

Create the set of reduced words that is the carrier set of \(G\). If Search is set to "DFS" (depth-first search) then words are enumerated in lexicographical order. If Search is set to "BFS" (breadth-first-search) then words are enumerated in lexicographical order for each individual length (i.e. in short-lex order). Depth-first-search is marginally quicker. Since the result is a set the words may not appear in the resultant set in the search order specified (although internally they will be enumerated in this order).

Seq(G, a, b): GrpRWS, RngIntElt, RngIntElt -> SeqEnum#

Search: MonStgElt                    Default: "DFS"

Create the sequence \(S\) of reduced words, \(w\), in \(G\) with \(a \le\) length\((w) \le b\). If Search is set to "DFS" (depth-first search) then words will appear in \(S\) in lexicographical order. If Search is set to "BFS" (breadth-first-search) then words will appear in \(S\) in lexicographical order for each individual length (i.e. in short-lex order). Depth-first-search is marginally quicker.

Seq(G): GrpRWS -> SeqEnum#

Search: MonStgElt                    Default: "DFS"

Create a sequence \(S\) of reduced words in the carrier set of \(G\). If Search is set to "DFS" (depth-first search) then words will appear in \(S\) in lexicographical order. If Search is set to "BFS" (breadth-first-search) then words will appear in \(S\) in lexicographical order for each individual length (i.e. in short-lex order). Depth-first-search is marginally quicker.

Example: Set (ex-6fc61b)#

We construct the group \(D_{22}\), together with a representative word from the group, a random word and a random word of length at most 5 from the group, and the set of elements of the group.

> FG<a,b,c,d,e,f> := FreeGroup(6);
> Q := quo< FG | a*c^-1*a^-1*d=1, b*f*b^-1*e^-1=1, c*e*c^-1*d^-1=1,
>                d*f^-1*d^-1*a=1, e*b*e^-1*a^-1=1, f*c^-1*f^-1*b^-1=1>;
> G<a,b,c,d,e,f> := RWSGroup(Q);
> Representative(G);
Id(G)
> Random(G);
b
> Random(G, 5);
a * d * b
> Set(G);
{ a * d * b, a * b, a * b * e, a * c, a * d, d * b, b * e, a * b * a,
  a * b * d, b * a, a * c * e, Id(G), b * d, c * e, e, f, a, a * e, b,
  c, a * f, d }
> Seq(G : Search := "DFS");
[ Id(G), a, a * b, a * b * a, a * b * d, a * b * e, a * c, a * c * e,
  a * d, a * d * b, a * e, a * f, b, b * a, b * d, b * e, c, c * e, d,
  d * b, e, f ]

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