The Value and Applications of Group Theory in Mathematics

[martinbn]

I don't understand! Why are you so worked up? The definition, you gave, is: Def: A group is a linear group if it is a subgroup of GL_n(F) for some field F. Then every finite groups is a linear group.

 

[fresh_42]

If students were asked in an exam to give an example for a linear group, and they answer $\mathbb{Z}_4$ or $A_4$ then this will get them no points. The theory of finite groups and the theory of linear (algebraic) groups are two different subjects. To mix them isn't helpful. In case we have a homomorphism $\mathbb{Z}_4 \to GL(V)$ it is a linear representation, faithful if the homomorphism is injective, but not a subgroup. Permutation matrices are a linear representation of (alternating) symmetric groups, they do not define the group.

And I want to see the theorem that states that every finite group has a faithful representation in a finite linear group, which is what you presented as common knowledge.

 

[mathwonk]

Pardon me for chiming in, but I suspect this is just a matter of differing use of language. Some of us say two groups are the same if they are isomorphic, which is of course a slight exaggeration. So some people consider a group to be linear if it is isomorphic to a subgroup of a general linear group. In the case of the symmetric group on n letters, one can let it act by permuting the column vectors of the identity matrix and get an isomorphic matrix group. I.e. a linear transformation is determined by its action on a basis, and is an isomorphism if it sends a basis to a basis, so we just need to permute the basis vectors. (One can use this as a trick for defining the sign of a permutation in terms of the determinant of its matrix.) Then since any finite group G of order n is isomorphic to a subgroup of the permutation group on n letters (by Cayley's theorem), G is also isomorphic to a subgroup of GL(n). This point of view on linear groups is taken e.g. in wikipedia, but of course not everyone need agree with that usage.

https://en.wikipedia.org/wiki/Linear_group

 

[Infrared]

And I want to see the theorem that states that every finite group has a faithful representation in a finite linear group, which is what you presented as common knowledge.

Every finite group is (isomorphic to) a subgroup of $S_n$ for some $n$ and $S_n$ has a faithful $n$-dimensional linear representation (permute the vectors in a given basis).

Edit: Mathwonk said this in his previous post

 

[martinbn]

Pardon me for chiming in, but I suspect this is just a matter of differing use of language. Some of us say two groups are the same if they are isomorphic, which is of course a slight exaggeration. So some people consider a group to be linear if it is isomorphic to a subgroup of a general linear group. In the case of the symmetric group on n letters, one can let it act by permuting the column vectors of the identity matrix and get an isomorphic matrix group. I.e. a linear transformation is determined by its action on a basis, and is an isomorphism if it sends a basis to a basis, so we just need to permute the basis vectors. (One can use this as a trick for defining the sign of a permutation in terms of the determinant of its matrix.) Then since any finite group G of order n is isomorphic to a subgroup of the permutation group on n letters (by Cayley's theorem), G is also isomorphic to a subgroup of GL(n). This point of view on linear groups is taken e.g. in wikipedia, but of course not everyone need agree with that usage.

https://en.wikipedia.org/wiki/Linear_group

This may be the case, but I don't think this was the issue. What about PSL? It is usually called a linear algebraic group although by definition it is not in GL, but a quotiont. I have never heard anyone call it isomorphic ot a linear group.

 

[martinbn]

Anyway, I am sorry to have derailed the thread. My comment was more about the fact that finite groups can be viewed as special cases of other classes of groups. They are topologycal groups, Lie groups, Algebraic groups, Affine algebraic groups and so on.

 

[WWGD]

How does Galois theory solve polynomial equations of fifth degree or over?Or does it solve mutivariable polynomials?Like in algebraic geometry?

I believe if the Galois group associated with the polynomial itself is solvable then there is a solution by radicals for the polynomial.

 

[mathwonk]

@martinbn: I am not sure what you mean by PSL, since you do not give the field. If you read the wikipedia link in my post you will see someone explicitly say that a certain case of PSL is isomorphic to a linear (matrix) group, via the adjoint representation.

Forgive me if i have misunderstood the issue, but to me if one distinguishes a matrix group, namely a subgroup of some GL(n), from a "linear group", namely one isomorphic to a matrix group, then as has been shown, ebvery finite group is linear in that sense. hence every PSL grouop over a finite field is linear, i.e. is isomorphic to a matrix group.

pardon me, best wishes.

 

[martinbn]

@martinbn: I am not sure what you mean by PSL, since you do not give the field. If you read the wikipedia link in my post you will see someone explicitly say that a certain case of PSL is isomorphic to a linear (matrix) group, via the adjoint representation.

Forgive me if i have misunderstood the issue, but to me if one distinguishes a matrix group, namely a subgroup of some GL(n), from a "linear group", namely one isomorphic to a matrix group, then as has been shown, ebvery finite group is linear in that sense. hence every PSL grouop over a finite field is linear, i.e. is isomorphic to a matrix group.

pardon me, best wishes.

I meant it as an algebraic group over any field. The point I was trying to make was that by definition it isn't a subgroup of the general linear group. But I have never seen anyone say something like "PSL is not a linear group, it is isomorphic ot one." In fact the usual definition of a linear algebraic group is a group which is an affine variety. Then the theorem: they are linear.

 

[FactChecker]

How can 1+...+1=0?it can't based on what we define as 1and 0.Could you tell me some examples of applications in number theory and if there are any applications of groups in string theory? I thought I read somewhere something related.I have not read till now about string theory except of some general things.

If '1' represents a 90-degree rotation and addition is defined as a sequence of rotations, then 1+1+1+1 = 0 because it gets you back to the original position. Abstract algebra and group theory applies to physical things and more than the number systems that are commonly used.

 

[trees and plants]

If '1' represents a 90-degree rotation and addition is defined as a sequence of rotations, then 1+1+1+1 = 0 because it gets you back to the original position. Abstract algebra and group theory applies to physical things and more than the number systems that are commonly used.

Thank you for the information about those fields of math that you gave me. What percentage of the results of those fields apply to physical things until today, do you know?

 

[fresh_42]

But I have never seen anyone say something like "PSL is not a linear group, it is isomorphic ot one."

Of course it is one. My only intention was to say, that those (linear algebraic, or projective) groups are usually infinite, and only of finite dimension, not of finite order. Of course there are some of finite order if the field is finite. But common meaning of the term finite groups are Galois groups rather than linear groups. The ordinary linear group is infinite, the finite ones are exceptions nobody really cares about, except in exams. The first line when it comes to linear groups is often: let us assume a field of characteristic not equal two. You never would assume such a thing when the topic is finite groups.

I got the impression that you confused dimension and order, or that at least readers might have gotten this impression.

 

[FactChecker]

Thank you for the information about those fields of math that you gave me. What percentage of the results of those fields apply to physical things until today, do you know?

A great deal of mathematics has been motivated in one way or another by the physical world. I would not be surprised if virtually all of group theory (and abstract algebra, in general) has applications in physics, chemistry, etc. For example, the regular pattern of crystal structures is rich with applications.