What is group theory all about?

[oldman]

A short answer to this question is perhaps "symmetry", or one might specify the properties that define a group (closure, associativity etc.). But are there less precise and more general ways of describing group theory that someone could point me to --- ways that perhaps better express the universal flavour of this subject? Something along the lines of "group theory is a way of classifying changes among the elements of a set of similar entities"?

It's such an abstract subject that I find it as difficult to describe as say, the counting numbers.

 

[CompuChip]

Sorry, but my answer will come down to 'symmetry' after all

For example, whether you want to count the number of ways in which you can color the sides of a cube, or you want to write down a theory of the universe, there will always be certain configurations that you want to consider as "being the same" because they are related by symmetry. If rotating the cube gives me something that looks exactly the same, I want to call it the same coloring; if a physical situation is reflection symmetric I want my theory not to change when I send \vec x \to -\vec x, etc.
Group theory is the mathematical framework that allows us to incorporate such symmetries in a fairly consistent and convenient way (and apart from that, of course, it is also interesting for pure mathematicians).

So if I'd have to make a general statement, after a quick moment's thought, it'd probably be something like:

Group theory is the theory that allows us to divide out symmetries from formal constructs
(that is, to intuitively identify formally different things)​

 

[oldman]

Sorry, but my answer will come down to 'symmetry' after all ... So if I'd have to make a general statement, after a quick moment's thought, it'd probably be something like:

Group theory is the theory that allows us to divide out symmetries from formal constructs
(that is, to intuitively identify formally different things)​

Thanks for the prompt answer. I have no quarrel with what you say. The reason I think one might look at a more general statement than one which includes the word symmetry (like your concluding sentence, for instance) is that I've been reading Mario Livio's popular book on group theory (rather floridly titled The Equation that Couldn't be Solved) and came across his example of a group that is made up of "all possible deformations that can be performed on a piece of" modelling clay (on page 164). This made me sit up and take notice, because I'd always thought of group theory and symmetry as being virtually synonymous. But there is nothing symmetric about such deformations! They're just changes, and I wondered if group theory is just a form of taxonomy of change. An astoundingly universal, clever, and important kind of taxonomy, of course, but nevertheless...

 

[jakthechemist]

So, this seemed like a good place to put this...

Greetings. This is my first post. I am teaching myself group theory so that I may better understand some of the concepts introduced in Quantum Mechanics.

From what I have found, there are 4 main axioms. I realize that an axiom is a statement that proves itself, but where did these 4 axioms come from, and why them? Why must a group be defined by them alone? Why not some other rule? I understand the axioms, I just don't understand why they were chosen or who chose them. I have found that 5th degree equations cannot be solved algebraically and that groups can be expressed as permutations, but no where can I find, in the plethora of mathmaticians that contributed to this theory, the reason for the axioms! >_<

If anyone has any thing to say that may shed some light on this I woiuld be most obliged!

 

[quasar987]

One can expect that the history behind how a set of axioms came to be universally accepted will be a complex one. What usually happens in mathematics is that someone (and often many people concurently) use certain new ideas to solve a given problem. And then these ideas or similar ones are used again to solve other problems, so they become recognized as important. Eventually, one tries to formulate a general theory from these ideas, and so lays down some tentative axioms. And then these axioms evolve, either because it is shown that some of the axioms are in fact redundant, or because the axioms are shown to be not general enough for comodity, or are too general for comodity.

 

[jakthechemist]

Ah yes, thank you, as I read more, I am finding just that to be the case. Every explanation seems to be 5 pages long listing at least 50 notable mathmaticians over the last 300 years. Apparently understanding how these axioms came to be will take more time; until I suppose I will have to take the at 'face value' as it were.

If anyone has any suggestions as to whose work would be a good starting place, it may save me some time, considering my background is nearly entirely based in chemistry. I was thinking of taking a look at the work of Able or Galois, but am unsure.

 

[HallsofIvy]

Historically, "groups" were created by Abel and Galois (well, maybe better to explain their results in simpler terms) and were "permutation" groups- the groups of possible permutations on a fixed number of symbols. It is relatively easy to prove that any group is isomorphic to a subgroup of a permutation group.