Does Group Theory tell the intrinsic nature of our universe?

Question

According to Group theory, a finite set of elements is a group (under certain operation) if closure law holds in it.

Now please let me give you two examples to elaborate my point.

Consider a finite set A = {2, 3, 4} under multiplication.
This set is not a group because closure law does not hold. So if I multiply any two elements of the set, I will basically create an element NOT in this set. 
E.g. 4 × 2 = 8 ∉ A

That is, Out of what I have with me, I can "CREATE" things which I don't have with me.

Now consider another set B = {1, ω, ω^2} under multiplication.
This set is a group because closure law holds. If I multiply any two elements of this set, I WILL NEVER BE ABLE TO CREATE ANY NEW THING.
E.g. 
ω × ω^2 = 1 ∈ B
1 × ω = ω ∈ B
1 × ω^2 = ω^2 ∈ B

So whatever multiplication I do within this set, I will never be able to create any new element out of what I have with me.

Our universe is pretty much the same. 
Assuming that there isn't an infinite mass or an infinite energy existing in the vicinity of space, we CAN NEVER create any new mass or any new energy out of what we already have with us. (E = mc^2 scenario is no exception either)

Now because our universe is finite in matter and energy just like any finite mathematical set, we cannot create any new matter/energy, no matter what mechanism of heat transfer we build.

Therefore, can we (in some philosophically abstract way) call our finite universe (if it is finite) A GROUP under the operation of "Mass-Energy Transfer"???

If yes, then I believe there must be some other operation which we can apply to the elements of this universe which would yield NEW energy/matter 'all in all'; just like in the set {1, ω, ω^2}, addition operation can create new elements.

If however, you are able to prove by some other way (i.e. inverse/identity) that our universe cannot be called a "Group" under this operation of "Mass-Energy Transfer", then we are sure that our universe has to be INFINITE in that case.

WHAT DO YOU THINK???

Ben · Accepted Answer

It might be interesting to apply group theory to the realm of mass-energy conservation, though I'm not sure that doing so would yield any results unknown about the day-to-day world.  You should know that many things have already come out of applying the notions group theory (along with further mathematical novelties) to the physical world.  For example, group theory plays an essential role in particle physics: the recent conjectures about "super-symmetry" come from looking at the ways that the patterns we observed might form part of a more vast, intricate, algebraic structure.

As for the specifics of what you said, you seem to have a misunderstanding about some aspects of group theory.  For example, just because you have a set that isn't closed under the group operation, doesn't mean you can generate infinitely many elements.  For example, if you take the elements {2,3} from the set of numbers under addition modulo 12, you can still only "create" the elements {0,1,2,...,11}.  Even worse, if you take the elements {2,4}, you can only "create" the elements {2,4,6,8,10,12}, missing half of the group.

Also, it's important to specify the process by which we create elements.  For example, take the integers under multiplication.  If you multiply any two integers, you get another.  However, the inverse of an integer is generally not an integer, which means that they don't form a group, and you can't get a group by "creating" elements the way you wanted to.  Also, with zero, we can't define an inverse even if we wanted to.

Finally, if you want to take some sort of mass-energy transfer as a group action, you assume that there exists a corresponding "inverse" action to bring us to the previous state.   However, the second law of thermodynamics tells us that most energy transfers are NOT reversible.

I think that's everything.  I hope that's given you something to think about.

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Does Group Theory tell the intrinsic nature of our universe?

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