# Intuition for groups

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.

• Could you clarify, are you having trouble gaining an intuitive feeling for groups, rings, fields, etc., just representation theory, or both? – Callus - Reinstate Monica Mar 10 '14 at 1:15
• Hello Callus, I would have to say both. The more I learn, i.e. representation theory, the less I seem to have a grip of what the view is. I can happily follow a proof, remember and understand results, answer questions and so forth, but don't really have a coherent connection between them, or any kind of easy way to see that this result is obviously really important, or this one is patently false. Apologies, I can't be more specific. – Daniel Johnston Mar 10 '14 at 1:31

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.