Intuition for groups

Question

This is quite a non-standard question, certainly for mathematics, though I believe it is no less important (for me and my peers, i.e. grads).

The course I am reading so far introduced us to Groups, rings, fields, etc. in the first year, progressing to characters, reps etc in the latter years.

However, even with representation theory I still don't feel like I have a good intuition.

How do you gain intuition in such an area. I know there will be a response of doing examples, but this doesn't really help - I've done lots already.

Am I doomed.

Answer 1

I'll take a stab at this. First off, intuition in mathematics is invaluable. That being said, it is often difficult to gain a level of comfort with abstract material on an intuitive level. However, any example/picture/ect. that can give some insight is wonderful. When dealing with groups, the first thing that comes to mind for myself is symmetry. A group structure indicates a degree of symmetry in an underlying object. I am quite certain you have seen groups being described as the symmetries of regular polyhedra ect. As for representations, they are in fact meant to make such symmetries more concrete. Instead of thinking of group elements as abstract symbols, we can obtain a set of matrices which behave the same way. Representations can actually be quite intuitive in some "nice" examples. For instance, a rotation group of a geometric figure can be visualized as matrices that take the figure to itself in the plane. However, when dealing with Hilbert spaces and vector spaces over arbitrary fields, this picture becomes less intuitive. I wish I had more insight as to visualizing such concepts (this lies outside of my typical area) but from experience with other abstract concepts such as higher dimensional manifolds, the best way to gain intuition is working with these things. You don't need to be able to draw what it "looks like" per se, all you need is to be able to draw a picture that means something to you. If a concept has meaning to you then its worth something (provided its accurate of course). An excellent series of videos on Youtube from a conference celebrating the proof of the Poincare conjecture has a lecture by Bill Thurston. It might be worth looking into (even though it doesn't mention representations) because Thurston was a true believer in intuition. Hope this helps a little.

Answer 2

A group is a set $G$ with an operation $\cdot$ that combines the elements of $G$ to give an element that also belongs to $G$. In order for $(G,\cdot)$ to be a group it must satisfy the following axioms: closure, associativity, identity element, inverse element.

For me the intuitive way to think about groups is to consider the Rubik's Cube group. This group consists of all possible states the cube can be in. All of the states can be reached by performing one of the face rotations: ${F, B, U, D, L, R}$ (Front, Back, Upper, Down, Left, Right) and these are called generators of the group. The operation $\cdot$ is the composition of those rotations. The closure of this group means that after performing a sequence of rotations, we end up with one of the possible Cube's permutations. Associativity is intuitive - if we perform moves $(F \cdot U)\cdot R$ we would end up in the same state as if we performed $F \cdot (U \cdot R)$. For each move there exists an opposite move (with the same face but in the opposite direction, e.g. $F$ and $F'$) that brings you back to the initial state (inverse element) and for each state of the cube there exists an identity element $e$ that means not moving the cube at all. Now you can abstract away the whole idea of cube and perform mathematical operations on this set using composition $\cdot$ and you know that this could be translated back to the physical orientation of the cube.

Sometimes different unrelated structures (atoms, particles) express some symmetry and therefore could be viewed as a subgroup of symmetry groups, e.g the Rubik's Cube group is the subgroup of the symmetric group $S_{48}$. This allows us to use common language to solve seemingly unrelated problems.

Answer 3

To be very intuitive, representation theory makes abstract algebraic objects "concrete" by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication respectively. Roughly, it makes abstract algebra into linear algebra, which is a concrete subject.