Talk:Group theory

Article Checklist for "Group theory"

Workgroup category or categories	Mathematics Workgroup [Categories OK]
Article status	Developing article: beyond a stub, but incomplete
Underlinked article?	No
Basic cleanup done?	Yes
Checklist last edited by	Greg Martin 00:27, 7 May 2007 (CDT)

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I have written quite a bit on groups, and it would be nice to have someone help make it more readable. I think the "examples" section looks a bit daunting to the eye, but I'm not sure how to organize it any better. - Jared Grubb 23:59, 3 May 2007 (CDT)

A few thoughts

It's worth noting that groups can be roughly divided into finite and infinite groups. The infinite groups may be discrete groups closely related to the finite ones (e.g. $\mathbb {Z}$), Lie groups, or much more complex groups. Some obvious examples of finite groups are:

1. (finite) cyclic groups
2. direct sums of cyclic groups
3. the symmetric groups and alternatiing groups
4. the dihedral groups
5. the unit quaternions

Beyond that, there are the "classical" groups which are the analogues of linear Lie groups over finite fields (e.g., $SL(n,\mathbb {F} _{q})$ and $PSL(n,\mathbb {F} _{q})$.

This article should also talk about representations of groups (i.e., homomorphisms $\rho :G\rightarrow GL(n,\mathbb {C} )$), and this would be an excellent place to mention that there are exactly 5 regular polyhedra. The complete classification of finite simple groups needs to be mentioned, too.

Other topics from group theory should probably include:

1. group actions
2. group presentations by generators and relations
3. the isomorphism theorems
4. the "Burnside" lemma (which is not due to Burnside, but the name is traditional)
5. the Sylow theorems
6. applications to Galois theory
7. Klein's Erlangen program (characterization of geometries in terms of the group of symmetries of the geometry)

It might be reasonable to talk about applications of group theory to classical and quantum mechanics, too. Greg Woodhouse 04:40, 4 May 2007 (CDT)

Great suggestions! I've created Talk:Group theory/Brainstorming so we can brainstorm about this topic: what needs to be here, what should be at Group (mathematics). - Jared Grubb 10:40, 4 May 2007 (CDT)

Definitions needed

Interesting topic. So group theory is how it was proven that quintics can't be solved. (If I knew that before I'd forgotten it.)

This is where you lose me: A solvable group, or a soluble group, is a group that has a normal series whose quotient groups are all abelian. Definitions are needed for: "normal series", "quotient group", "alternating subgroup", "$A_{5}$", "symmetric group".

In this sentence: A free group is a group in which every element of the group is a unique product, or string, of elements of some subset of the group. It needs to be clarified whether "unique" means that each element can be expressed as a product in only one way (up to use of the identity element, presumably), or whether it means that a given string can only represent one element (obviously true given the definition of binary operation) or perhaps that a given subset can only represent one element regardless of which order they're put into a string. --Catherine Woodgold 20:25, 5 May 2007 (CDT)

Those definitions are incomplete in this text, but I did that on purpose. I'm worried that if we put full-fledged definitions for every concept mentioned into this article, it will get large and unwieldy. So my intention is that the articles solvable group, quotient group, etc. would each discuss those concepts in greater detail, whereas this article just mentions enough to explain the role of each concept in group theory. What do you think of that? - Jared Grubb 10:45, 6 May 2007 (CDT)

That may be OK, but if a term such as "normal series", "quotient group" etc. is used, then in my opinion it needs either an abbreviated informal definition, or a full definition, or a link to where the definition can be found. For now, they could just be red links if there is going to be a page about that concept later -- but remember that we don't create pages just to put definitions (it's not a dictionary). Another option is to use footnotes. --Catherine Woodgold 11:17, 6 May 2007 (CDT)

I have added some footnotes, and there is a section about quotient groups (although the placement is a bit suboptimal since we talk about quotient groups before we define them, but I hate to bring that section up any higher....). I am planning on rewriting the "Examples" section, and I think we can define S5 and A5 there. Read it through and let me know what you think. - Jared Grubb 12:21, 6 May 2007 (CDT)

Think about it this way. If you adjoin all the roots of an irreducible polynomial to a field, you get an extension called the splitting field of the polynomial. For example, you get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{C}} from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} by adjoining the roots of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 + 1} and all their linear combinations, of course. Field extensions that arise this way are called normal extensions. Now, in charactedristic 0, these polynomials won't have repeated roots in the splitting field (this is expressed by saying that the extension is separable), so in the splitting field we can write

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = (x - \alpha_1)(x - \alpha_2) \cdots (x - \alpha_n)}

Now, the coefficients of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} are in the base field, so any automorphism of ther splitting field that fixes the base field, must leave the polynomial intact, and so must permute the roots. (This group is known as the Galois group of the extension, and I think it was Galoi who first used the word group.) This gives us a permutation representaton (group action) of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ \alpha_1, \ldots , \alpha_n \}} . Recall that the alternating group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_5} is simple, so the permutation group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_5} cannot be solvable, meaning there is no sequence of normal subgroups

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_k = G}

such that the quotient groups Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_i/G_{i - 1}} are abelian. But who cares? Well, it turns out that if K is a normal extension of k and L is an intermediate field, such that K/L is a normal extension, Gal(K/L) is a normal subgroup of Gal(K/k) and conversely. But what does it mean to solve an equation by radicals? Basically, the only extensions we can take are splitting fields of polynomials of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^k - a} , and they have abelian Galois groups, meaning, in turn, that the full Galois group must be solvable. If we can find a polynomial with a splitting field having Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_5} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_5} as Galois group, we will have shown that not every polynomial can equation can be solved by radicals.Greg Woodhouse 12:08, 6 May 2007 (CDT)

Was that a volunteer to write quintic polynomial? That's what I heard... :) - Jared Grubb 12:21, 6 May 2007 (CDT)

Narrative

Anyone want to now take a stab at superimposing a narrative structure onto the set of facts currently included? - Greg Martin 00:27, 7 May 2007 (CDT)

We could perhaps start out by noting (perhaps in different words) that the mathematical notion of a group abstracts from the more intuitive notion of symmetry. This is easy to picture in the case of Lie groups, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle SO(3)} is the group of rotational symmetries of 3-space or, more boringly, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle SO(2)} is the group rotational symmetries of the plane (i.e., the circle group). The dihedral group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_4} is a compelling example, because it is just the 8 symmetries of a square (4 rotations, reflection is the x-axis, the y-axis, and each of the two diagonals). Someone more gifted than me with graphics might include pictures of the five regular polyhedra (tetrahedron, cube, octahedron, dodecahedron and icosohedron) and note that there are no others, in contrast with the two dimensional case where there is a regular n-sided polygon for each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n > 2} . Why is this? By deMoivre's theorem, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_n} can be embedded in the circle group by choosing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{2\pi / n}} as a genertator. Unfortunately, analysis of the finite subgroups of SO(3) isn't so easy, and I don't know any easy way of showing why there should only be the five (or four, actually - symmetries of the cube and the octahedron are the same). Another accessible motivating example is Klein's Erlangen program, in which geometries (Euclidean, projective, spherical, hyperbolic, etc.) are described in terms of the group that leaves invariant those properties we wish to describe a "geometrical" (thus, 3-dimensional Euclidean geometry amounts to the study of the action of the semidirect product of SO(3) and the translation group - the additive group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbb{R}^3, +)} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^3} .

Of course, the Galois group is a symmetry group, too, though in this case what is left invariant is the equation(s) defining the field extension. I'm less convinced that this is a useful motivating example, but it is the motivation for the abstract concept of a group. So far as I know, Évariste Galois was th first person to actually use the term group.) Greg Woodhouse 03:20, 7 May 2007 (CDT)