# It's a Group. So what?

## Main Question or Discussion Point

I was just wondering why is it so important for certain tools used in physics and engineering to form a group? Like a vector space for example. What advantage is there if it forms a group?

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The useful point about mathematical groups is that you know how many members any given group has.
So if your property of interest is a member of a known group you can tell how many other members there are and if you have missed one out in you analysis or if there are any more to look for.

Finite groups are used extensively in things like crystallography.
E.g. all of the physical, regular crystals that can appear in nature must be finite subgroups of the rotation group. This allows for their complete classification.

Also Lie groups are indispensable.

Changes in your point of view of a system (i.e. Galilean, special and general relativity) must form a group. So that you can always make two changes to get another. You can always undo a change etc...

In relativistic field theory, if the field is to be obey special relativity, then it must be a representation of the Poincare group. If the field is to be fundamental (not phenomenological) then the representation must be irreducible. If we add quantum mechanics then it must be a unitary rep. If there are other fundamental (eg gauge/flavor) symmetries in the theory, then the field must also be an irrep of those.
This then gives us the classification of particles in quantum field theory.
http://en.wikipedia.org/wiki/Particle_physics_and_representation_theory

Then you get to technical issues, such as if you want a vector field in quantum field theory, then it must be a gauge field for a reductive (semisimple + U(1) factors) group.

AlephZero
Homework Helper
The one-word answer is "symmetry". (Note, the word "symmetry" includes some very subtle concepts, as well as the obvious ones).

Two more cents: a group allows for cancellation. For example, if you go on adding vectors, opposite vectors cancel out, and you can forget them and sum just the others. Similarly, if some vectors form a loop (add up to zero), you just take them out and sum only the others. All this springs from the group axioms.

Not to mention that when we can describe something as a group (or a Ring, Field, Algebra, etc) then there are many tools we have at our disposal to figure stuff out about a particular group/ring/etc.

So, a vector space forms a group. If you tell me this, I automatically know a lot of stuff about it. I know that you have defined some operation on it. I know that this operation has inverses. I know there is an identity element. I know that inverses are unique. I know that I can form subgroups. I know that if T is a homomorphism from V into itself then the kernel is a subgroup. I know that if the kernel is just {0} then the T is an isomorphism.

But, just wait till you get to higher algebra classes. You will see that a vector space, say R^3 forms something called an algebra. Then you can see that matrcies and linear transformations are all the same (well isomorphic.)

Also, a vector space is an algebraic structure in its own right. In fact, some texts define a vector space to be a group over a field of scalars. So, why do we characterize things as vector spaces? Well, we know a whole lot about vector spaces. Same with groups.