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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 ≤ length(w) ≤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 ≤ length(w) ≤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 GrpRWS_Set (H74E9)

We construct the group D22, 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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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013