How to properly understand finite group theory

[Terrell]

I do have a fair amount of visual/geometric understanding of groups, but when I start solving problems I always wind up relying on my algebraic intuition, i.e. experience with forms of symbolic expression that arise from theorems, definitions, and brute symbolic manipulation. I even came up with a "physical" (far from perfect) interpretation of the first isomorphism theorem. But when I get to solving problems, it's as if this intuition might as well gets thrown out the window because it's almost useless. It's really frustrating . Of course, I could not solve all the problems, but there are problems I could successfully solve and have no idea what I'm writing. Is this perfectly normal?

 

[member 587159]

I do have a fair amount of visual/geometric understanding of groups, but when I start solving problems I always wind up relying on my algebraic intuition, i.e. experience with forms of symbolic expression that arise from theorems, definitions, and brute symbolic manipulation. I even came up with a "physical" (far from perfect) interpretation of the first isomorphism theorem. But when I get to solving problems, it's as if this intuition might as well gets thrown out the window because it's almost useless. It's really frustrating . Of course, I could not solve all the problems, but there are problems I could successfully solve and have no idea what I'm writing. Is this perfectly normal?

Give it some time. You might develop some form of abstract intuition.

 

[Terrell]

Give it some time. You might develop some form of abstract intuition.

what is this abstract intuition? Do you mind illustrating some basic examples of this? thank you.

 

[fresh_42]

I do have a fair amount of visual/geometric understanding of groups, but when I start solving problems I always wind up relying on my algebraic intuition, i.e. experience with forms of symbolic expression that arise from theorems, definitions, and brute symbolic manipulation.

This is one possible way of imagination. It helps for problems like the following: "Definition 1 is given by (associative, neutral, inverse) and definition 2 is given by ($a\cdot x = b$ is solvable). Show their equivalence." Here the algebraic point of view fits best.

Other ways are: geometric (e.g. orthogonal groups), functional (e.g. automorphisms, derivations) , analytic (e.g. Lie groups). Each of them has its legitimacy, depending on the context. What they have in common is the interpretation as elements doing something on an object: solving equations, leaving angles constant, conserving the structure resp. being a differential, or containing smooth paths among their elements.

I don't think there is a one fits all intuition. If you consider e.g. $GL(3,\mathbb{F}_7)$ you will be stuck with combinatorics, one way or the other. If you consider semi-direct products, then you will have to combine your algebraic point of view with an automorphism and its action on certain group elements. If you study crystallography, then you will have a hard time without a geometric intuition. But to study Galois theory, a geometric intuition likely won't help you a lot. In this sense

Give it some time. You might develop some form of abstract intuition.

is probably the only possible advice, in the meaning that certain intuitions fit in certain situations and the more you study the latter, the better will become the former.

 

[member 587159]

what is this abstract intuition? Do you mind illustrating some basic examples of this? thank you.

Abstract intuition for me means that when you see a problem and your instinct immediately tells you: "this theorem might be useful", or "this strategy" might work.

In a certain sense, it means that you have become familiar with the theory, and you know the common tricks.

It's a little bit hard to explain.

 

[Terrell]

I don't think there is a one fits all intuition.

I think I will have to agree with you here. This eased some self-doubts. Thanks!

Abstract intuition for me means that when you see a problem and your instinct immediately tells you: "this theorem might be useful", or "this strategy" might work.

I do get some of these. Maybe I am just feeling a bit too impatient.

 

[fresh_42]

I think I will have to agree with you here. This eased some self-doubts. Thanks!

Since I also have difficulties with imaginations aside the algebraic one, I might be the wrong defender of other views in this case. What comforts me is, that whatever a group is supposed to represent, the algebraic view will always provide me with something I can calculate with.

E.g. I frequently look up the dihedral group, since it occurs in posts here. It's a good example for your question, because we have the following choices:

• $D_{n}$ is the isometry group of an n-gon.
• $D_{n}$ contains all rotations and reflections of an n-gon, i.e. congruences.
  
• $D_{n}$ contains a set of $(2\times 2)-$matrices with trigonometric entries, i.e. $\sin \alpha_k\; , \;\cos \alpha_k$.
  
• $D_{n}$ is a certain subgroup of $\operatorname{Sym}(n)$, i.e. a permuation group.
  
• $D_{n} = \langle r,s\,|\,r^n=s^2=srsr\rangle$
  
• $D_{n}$ is a set a functions which can be (and were) used as an alternative to modulo checksums, e.g. on German DM bills.

Now make your choice! You can see, that the area you come from probably determines the preferred representation.

 

[lavinia]

You might like to experiment with matrix groups and see what you come up with. Try matrices with only ones, minus ones, and zeros and which permute the standard basis in Euclidean space. All of these are symmetries of Euclidean space.

For instance what is the structure of the group generated by $\begin{pmatrix} 0001\\1000\\0100\\0010 \end{pmatrix}$ and $\begin{pmatrix} 0100\\1000\\0001\\0010\end{pmatrix}$ ?

What if the first matrix is changed to $\begin{pmatrix} 000^{-}1\\1000\\0100\\0010 \end{pmatrix}$

For example you might like to find the normal subgroups and examine the quotient groups. This will give a picture of how the group is constructed from the normal subgroup and the quotient group. Different normal subgroups give different pictures.

Groups can also be thought of as symmetries of themselves. One way to do this is to let the group act on itself by conjugation.

 

[WWGD]

Abstract intuition for me means that when you see a problem and your instinct immediately tells you: "this theorem might be useful", or "this strategy" might work.

In a certain sense, it means that you have become familiar with the theory, and you know the common tricks.

It's a little bit hard to explain.

Maybe it comes down to Terrell having reached the Pareto 20-80. Sorry to say it but there is a kind of steep hill from there on. Still, if you have good intuition, you're in good company with, e.g, chaos' Benoit Mandelbrot.