Understanding Group Structures: A Scientist's Perspective

[modnarandom]

What does it mean to give a group structure? I'm working on a problem and part of it asks for the structure of the group. The law of composition and generators seem to be given already (and an expression that says that a^2 = 1 for any elt a of the group). Is there anything to do other than verifying the group axioms?

 

[Benn]

Is the group abelian?

 

[modnarandom]

Nope. Also, the question specifically asked to "compute" the group structure (wasn't sure what that meant).

 

[chiro]

Hey modnarandom.

One of the important points of a group is that can consider processes that are invertible. The idea of invertibility is critical when studying a system especially when you are looking at systems that "evolve".

Think of a chess game: you have a situation where all the pieces can not only move in their own ways, but every single action is invertible. Trying to understand a system like a chess game in an abstract manner requires one to form something like a group. If mathematicians find very general statements about groups that categorize general behaviour, then this behaviour can be applied to any kind of group system (like a chess game).

There are other reasons that have to do with symmetry as well.

Mathematics pretty much tries to take really general things and make sense of them in a way that provides insight. It's a lot harder to do this on the abstract level because the things you are looking at are not specific. When things are more constrained they are a lot easier to analyze, but when they are abstract it becomes a lot harder because you have to choose a way to classify and the space for the choice becomes a lot larger and it means you need techniques that are a lot different than if you were analyzing something very specific.

 

[Benn]

The group is abelian if the each element a in the group satisfies a²=1 (prove this). The more structure you give something, the more you know about it. Commutativity,closure, etc are examples of structure.

 

[rasmhop]

Could you perhaps tell us the actual example. Structure can mean slightly different things to different people in different contexts. It can mean high level concepts such as whether the group is Abelian (as Benn indicates a^2 = 1 implies that it must be), but in my experience this is not what is meant when one asks for THE structure rather than some structure.

The group structure of a group usually means its composition operation (i.e. a multiplication table). Thus if you already know a complete description of composition, then I would say you know the structure.

However, usually we would like the group structure in some nice form. This means that rather than writing a complete multiplication table:
ab= ba^2
a^3 = b^2
cab = aba
...
we would prefer to identify our group with some well-known group such as \mathbb{Z}, C_n (cyclic group of order n) or D_{2n} (dihedral group of order 2n).

Given the information you have provided I would guess the problem at hand is to determine the group structure of something like the group G generated by elements a,b with relations
x^2 =1 for all x in G
In that case you should be able to identify G with either the Klein four group or the cyclic group of order 4 (figure out for yourself which one if this is indeed the problem). That is how you would determine the group structure of G, by saying which group it is isomorphic to.

 

[lavinia]

What does it mean to give a group structure? I'm working on a problem and part of it asks for the structure of the group. The law of composition and generators seem to be given already (and an expression that says that a^2 = 1 for any elt a of the group). Is there anything to do other than verifying the group axioms?

I would guess that you are being asked to identify the group with some known group e.g. show that the group is a dihedral group of order 8, i.e. some group that you already know.

 

[modnarandom]

@Benn, rasmhop: Oops! Yes, I proved that the group is actually abelian.
The problem is on p. 80 of http://www.math.harvard.edu/hcmr/issues/6.pdf
(actually working on problem A11-6, thinking of submitting at some point - so I just want clarification of the problem). Thanks for the help! I think I found the group structure (showed it was isomorphic to some group).