       Re: groups

• To: mathgroup at smc.vnet.net
• Subject: [mg7527] Re: groups
• From: Chun Che Li <ccli at math.ucla.edu>
• Date: Tue, 10 Jun 1997 23:18:03 -0400 (EDT)
• Organization: University of California, Los Angeles
• Sender: owner-wri-mathgroup at wolfram.com

```Hi! Hope this program can help you.
-------------------------------------------------------------
(* Multiplication and Inverse for permutation group *)
Mult[a_List, b_List] := Module[{n = Length[a]}, Table[a[[b[[i]]]], {i, 1,
n}]]
Inv[c_List]:=Last[Transpose[Union[Table[{c[[i]],i},{i,1,Length[c]}]]]]

(* generators -> unit, generators, and inverse of generators *)
Initiate[generators_List]:=
( Unit=Range[Length[generators[]]];
Union[Join[generators,{Unit},Inv/@generators]])

(* elements -> elements multiply elements , till this set remain unchange *)
Group[generators_List]:=
Module[{Elem=Initiate[generators],NoOfElem=0},
While[NoOfElem=!=Length[Elem],NoOfElem=Length[Elem];
Elem=Union[Flatten[Outer[Mult,Elem,Elem,1,1],1]]];
Elem]
______________________________________________________________________
Group[generators] gives the group generated by the elements.

e.g.
In:=Group[{{2,3,4,5,1}}]
Out:={{1,2,3,4,5},{2,3,4,5,1},{3,4,5,1,2},{4,5,1,2,3},{5,1,2,3,4}}

In:=Group[{{2,3,4,5,1},{2,1,3,4,5}}];
In:=Length[%]
Out=120

This show that the group S_5 is generator by (23451) and (12)

Charles,Li Chun Che

On 4 Jun 1997, Will Self wrote:

> I would like a function that accepts as inputs some permutations of
> {1, 2, 3, . . . , n} and returns the group generated by them.
>
>
> Will Self
> Montana
>
>
>

```

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