Re: Groups defined by as simple as possible rules... as computer strings?



Anton Suchaneck wrote:
Snip
> Of course sometimes these defining equations contain more than one element
> and it would be good to minimize the number of elements used.
>
> Is there a method to do this systematically and is it possible to use
> equations of the form "...=1" only? (say a*b*c=1 and b*b=1)
>
> I'd find it instructive to imagine groups as computer strings (e.g. "gg",
> "ggg") which can be catenated with rules of cancellation (e.g.
> "gg"+"ggg"="ggggg"="g" since "gggg"=""). In this model the associative
> property is implicit.
Snip
> Anton
Other people have explained some of the theory. I tackled it as a
computer problem.
Different ways of generating groups (together with quasigroups,
loops, and signed tables) are demonstrated in the database "loops" in
the package GroupMLHoop, available from
http://library.wolfram.com/infocenter/MathSource/4894. This contains
prescriptions for almost all small finite groups with up to 73
elements, generating "protoloop" preferred isomorphs as Cayley
multiplication tables. The resulting tables may be "index tables", in
which elements are represented by indices {1,..i,,,,m}. In many cases,
a rule ca[m,k]:=Mod[i+If[EvenQ[i],j k-k+1,j]-1,m,1] gives the entries
"ij"; different values of k give several classes of group. E.g. ca[6,5]
gives D3.
Index tables can be "composed" to give larger tables, using direct or
indirect composition. D3 is the indirect composition of C2 & C3.
The elements may be "words", lex-ordered strings of generators such as
"aab"; each generator is a "root of unity" (as you imply) with rewrite
rules (relations) such as "ba"->"aab" that recover lex-order from
products obtained by string concatenation. These are needed for all but
the Abelian groups. ge[{3,2},{"ba"->"a2b"}] also gives D3 by specifying
a^3=b^2=1.
The elements may be "symbols", {a,b,...}. These can represent unitary
monomial matrices, with matrix multiplication as the operation. The
matrices {{0,-b},{b,0}} and {{a,0},{0,a^2}} create D3.
The elements may represent permutations (every group can be defined in
terms of permutations) but this is relatively clumsy. sgroup[n] creates
Sn via permutations.
Procedures to create generalized Clifford and Cayley-Dickson algebras
create "signed tables" that can be expanded to loops which may be
groups.

None of these procedures satisfies your desiderata; GAP
http://www-gap.dcs.st-and.ac./uk/~gap may provide defining permutations
(I have not looked for them). My preferred procedure is via generators
and relations. I have only failed to find generators (for groups with
less than 64 elements) in one case, GAP[SmallGroup(48,28). But, as
others have pointed out, there is no solution to the "word problem".
Have fun.
Roger Beresford.
"Studious of elegance and ease, myself alone I seek to please." (John
Gay.)

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