Determining Defining Relations for a Group

wnorman27

1. The problem statement, all variables and given/known data
Given some group G with generators g[itex]_{1}[/itex],g[itex]_{2}[/itex],...,g[itex]_{n}[/itex] as well as a description of the action of g on the elements of some set S={s[itex]_{1}[/itex],s[itex]_{2}[/itex],...,s[itex]_{k}[/itex]}, how in general does one go about finding a complete defining relations (and showing they are complete)?


2. Relevant equations
For example, the group of symmetries of an n-gon has generators R (rotation) and D (flip across a diagonal) with defining relations R[itex]^{n}[/itex]=1=D[itex]^{2}[/itex] and RD=DR[itex]^{n-1}[/itex]. Both R and D can be described by their effect on the vertices of the n-gon.


3. The attempt at a solution
I understand that the example I gave has a complete set of relations, and it seems to be something of a requirement to show the orders of the generators (if finite) and how any two generators "commute" with one another. Is this at all on track? Is there some general method?

algebrat

I don't recall a rule of thumb. It can be a problem to figure out if you have enough relations to make a group finite, or to get the intended group, or that you did not add so many relations that you have the trivial group. Sometimes, it is easier to prove which group you have by representing it in the symmetric group, or with matrices, or geometrically. Then once you have existence, you may be able to set up an isomorphism with the generator/relation presentation.

The way two elements "commute" with one another, as you mentioned above, might help in general since this would help write any "word" as a power of x times a power of y. I'm implicitly imagining a group.