Help in understanding groups (undergraduate level).

[AAQIB IQBAL]

I have studied a fair portion of groups, but couldn't imagine what they are all about. Please help me in this regards.

 

[Stephen Tashi]

Here's one way to think of them in a concrete manner:

Think of a set of functions S such that each function in S has domain D and also range D.

Then if f and g are functions in S, the composition f(g(x)) also has domain D and range D. But it might not be a function in S.

So let's define a "better" kind of set of functions. Suppose H is a set of functions, each of which has domain D and range D. Further suppose that if f and g are functions in H then f(g(x)) and g(f(x)) are also functions in H. The composition of functions is a kind of "multiplication" of the things in H. It is associative since f(gh) = f(g(h(x)) = (fg)h = f(g(h(x))). If we include the identity function I(x)= x in H, then the multiplication has something that behaves like 1 since I(f(x)) = f(x) = f(I(x)).

One think lacking about the "multiplication" in H is that there may not be any multiplicative inverses. So let's define an even better kind of set of functions G with these properties: Each function in G has domain D and range D. If f and G are functions in G then f(g(x)) and g(f(x)) are functions in G. The identity function I(x) is an element of G. If f is a function of G then the inverse function of f is in G.

With that definition, the functions of G must be 1 to 1 since they have inverses. The composition of functions is analagous to ordinary multiplication with the exception that it is not commutative. f(g(x)) may not be the same function as g(f(x)).

 

[lpetrich]

You can think of groups as generalizing addition and multiplication.

If you want something to picture, try rotations and reflections. Like rotate or reflect a regular polygon in a way that makes it look the same afterward. You'll see an important application of group theory: symmetries.

For the smallest nonabelian (noncommutative) group, try rotating and reflecting a regular triangle and seeing what happens to the vertices. You can do that with other polygons also.

I suggest that you invest in some computer-algebra package like Mathematica or Maple or Maxima or Sage, or failing that, some numerical package like Matlab or Octave. See if you can program it to do various manipulations of small groups.

 

[quasar987]

Groups are a very basic type of structure one can put on a set. Namely, a binary operation satisfying a couple of axioms. Structures are the weapons of the mathematician. This justifies studying abstract group theory for its own sake. For instance, the mathematician one day might encounter a certain set that he dearly wishes to understand better. And he'd say "If only I could but a group structure on this set, then life would be good because I know all about groups."

This is one point of view of what group theory is about. There are others.

 

[Tac-Tics]

Groups are about doing stuff that you can undo. And one of the things you can do is to do nothing.

The group of addition over the integers. You can add an integer to what you have, you can subtract it (which is how you undo stuff), and you can add 0 (which is how you do nothing).

The group of multiplication over the positive reals. You can multiply a real to what you have (doing stuff), you can divide by any real (undo it), or you can multiply by 1 (do nothing).

The group of walking in a direction. You can pick a direction and a distance and walk that far (doing stuff), you can pick a direction and a distance and walk the OPPOSITE direction that distance (undoing stuff), or you can just stand there (do nothing).

The group of invertible matrixes under multiplication. You can multiply by a matrix (do), multiply by its inverse (undo), or multiply by its identity (do nothing).

The group of invertible matrices also allows us to talk about groups of rotations, translations, skews, etc. Very useful in science.

The group of invertible functions under composition. You can apply f (do), you can apply f^-1 (undo) or you can apply the identity function (do nothing).

The rubicks cube group allows you to rotate any block (do), rotate it the other way (undo) or just let your cube sit there because you couldn't solve it (do nothing).

The permutation groups allow you to swap positions of two elements in a list (which is both the do and the undo), or swap an element with itself (do nothing).

 

[HallsofIvy]

I will also point out that groups, historically, originated as "permutation" groups, looking at the way permutations can be combined. This became important as a result of Galois' proof that there cannot be a formula (in terms of roots) for the zeros of a polynomial of degree 5 or better. He worked with formulas for permuting the zeros of such polynomials.

It can be shown, fairly simply, that every group of order n is isomorphic to a subgroup of the group of permutations on n objects.