# Anyone know of a good website that reviews group theory?

1. Oct 18, 2011

### LogicX

I have never taken an abstract algebra course. So now that in inorganic chemistry they are throwing group theory at us, I am really confused. It all seems like a jumbled mess of random relationships and numbers.

I am fine with point groups. Symmetry comes easily to me, so that is not an issue at all. I just need a website that goes over the basic of group theory and how you use it in chemistry. My textbook is a joke; it has about 5 pages on the background of group theory and that's it. My in class notes are difficult to follow for this subject, although I suspect they will end up being my main source of studying.

So, does anyone know of such a website? There are a ton of sites out there reviewing symmetry and point groups but I already understand those. I need the step further of using them in basic group theory.

2. Nov 5, 2011

### Sprooth

In order to gain a better understanding of groups and to feel more comfortable with them, it is important to understand the idea behind groups and to look at a number of specific examples. The main objective of algebra is solving equations. We have systems of numbers such as integers, rational numbers, real numbers, and we have operators to create equations between these numbers, then we develop methods of solving these equations. What properties of these systems makes it possible to solve equations? Over many decades brilliant mathematicians boiled down the most important properties of these systems of numbers, and hence we have the three basic axioms of group theory.

Algebra has been described as the study of structure. Solving equations is essentially using the overarching structure of groups to unravel the smaller structure of specific equations. Abstract algebra gets its name from the fact that we "abstract" away the extraneous information and focus on only the essential properties.

I would recommend looking at specific examples such as: the integers, the real numbers, the cyclic group of order n, and the dihedral group of degree n. Try verifying the three axioms for each case, and imagine what it means for these axioms to be satisfied. Probably the integers and real numbers will be most familiar, and will most readily fit your conceptions of structure, but I imagine the latter two finite groups will begin to shape your understanding in the direction it will need to go for chemistry, where the cyclic properties will become prominent.

I usually find the planetmath articles to be good. If I were you, I would look through them or search for things like "basics of group theory" or "simple group theory examples".

Last edited: Nov 5, 2011
3. Nov 5, 2011

### Sprooth

I quickly found this website. It seemed pretty informative and covers what I'd consider a good range of concepts to introduce you to group theory.
http://dogschool.tripod.com/index.html

Another note: sometimes the "axioms of group theory" are considered to be four or perhaps five in number. When I referenced the "three axioms", I intended for the other one or two to be included implicitly.
(0) A group G is a set of elements (usually called g, h, etc.) along with an operation * between these elements. The operation is associative but not necessarily commutative. G satisfies these axioms:
(1) For any two elements g, h in G we have that g*h is also an element of G,
(2) There is an identity element e in G such that g*e = e*g = g, for all g in G,
(3) For each element g in G, there is an inverse element h in G such that g*h = h*g = e. We usually denote the inverse of g by g^-1.
The associativity of the operation * is often considered an axiom, and one could conceivably consider other parts of (0) to be axioms as well.

I think if you put time into understand group theory, you will find it rewarding. The more I study algebra, the more I become fascinated with it's ubiquity and elegance.

Last edited by a moderator: Apr 26, 2017
4. Nov 6, 2011

### homeomorphic

Website?

I generally prefer the library to websites, unless I just want to look up one or two things, although there is more and more stuff available online.

Here's a textbook.

http://www.math.uiuc.edu/~r-ash/Algebra.html

I'm not sure what you mean by "point groups"? Maybe that's what we would call permutation groups?

5. Nov 7, 2011

### Deveno

as i understand it, in inorganic chemistry they classify point groups (what we would ordinarily call symmetry groups of geometric figures) according to character tables.

so really, it's not "groups" per se, but group representation theory as symmetry operations (in R2, R3, usually). it's the back-door into group theory, and i imagine many chemistry students find it totally bewildering.

so it's "more" than just group theory, to have it all come "full circle" for you, you'll need to not only understand group theory, but a decent amount of linear algebra, as well (the idea behind all this, is to turn "abstract" groups, into "concrete things" we understand better, namely, linear transformations).

for example, the symmetry group of H2O, has 4 elements:

E = the identity
C2 = a rotation of 180 degrees about the z-axis
σv = reflection in the xz-plane
σv' = reflection in the yz-plane

but an algebraist would just say "the Klein 4-group (or Viergruppe)", perhaps as:

{e,a,b,ab} a2 = b2 = e, ab = ba

or perhaps as Z2xZ2 = {(0,0),(1,0),(0,1),(1,1)}.

perhaps a more familar way to look at this group is the set of 4 3x3 matrices:

$$\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix},\ \begin{bmatrix}-1&0&0\\0&-1&0\\0&0&1\end{bmatrix},\ \begin{bmatrix}-1&0&0\\0&1&0\\0&0&1\end{bmatrix},\ \begin{bmatrix}1&0&0\\0&-1&0\\0&0&1\end{bmatrix}$$

although a close inspection will reveal all the action is happening in the 2x2 matrix blocks up in the upper left corner, the z-coordinate is "just along for the ride" (this representation isn't irreducible).

the general idea is this: if you have a set X of "somethings", and you have a collection of "transformations" (functions) that do something to X, and leave it "unchanged" (i am being deliberately vague here, on purpose) we call the set of such transformations symmetries. the most general set of such functions even has a special name: SX, or Sym(X), and is called the full symmetric group on X. this is sort of "the big daddy group" where you can find most of the "baby groups" (this is a silly way of saying something that has it's own name, Cayley's theorem).

so (all of the proper and correct axioms stated in other posts aside), a group is (at its heart) a collection of "reversible transformations of a set". in chemistry, the "set" in question is a collection of atoms (a molecule), and the transformations are geometric operations that give you a molecule "you can't tell from the original" (hence the name "symmetry").

a proper course in group theory, will spend a lot of time on studying small sets first (there is a lot of richness, even with as few as, say 4 elements, in the set X). you'll want to learn about:

cyclic groups
dihedral groups
the symmetric group Sn (this one is a doozy)
general linear groups <--this, especially, is important
special linear groups
orthogonal groups
special orthogonal groups

because these will be the ones most useful to you.

learning how character tables are created, and what they mean, is a bit beyond what you will find in most books on group theory, even good ones. after (and if) you learn a bit more group theory (i like the dogschool site linked to above, it's easy on a newbie), if you're feeling more ambitious, you can try:

but i must warn you, it's not for the faint of heart (it's a good read for physics students, too).

6. Dec 12, 2011

### mathwonk

there are basically two standard kinds of groups, symmetric groups, and linear groups, i.e. (finite) groups of permutations of finite sets, and (infinite) groups of linear transformations of finite dimensional vector spaces (matrix groups).

so check out GLn (all nxn matrices of determinant ≠ 0), On (distance preserving linear maps), SLn (volume preserving and orientation preserving linear maps) and Spn, (these preserve a "symplectic" form, e.g,. the intersection product of curves on a compact surface).

7. Feb 1, 2012

### mal4mac

8. Mar 18, 2012

### Cod

9. Mar 18, 2012

### DIOGYK

10. Mar 19, 2012

### Daniiel

11. May 20, 2012

### SuperiorPrep

12. Dec 2, 2013

### DrDu

Last edited by a moderator: May 6, 2017
13. Dec 31, 2013

### MikeJohnson

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14. Jan 1, 2014

### djh101

Well, there already appear to be a good amount of responses, but I would still like to chime in with my recommendation: Symmetry by McWeeny (sorry that it's a book rather than a website). This really is an excellent book on group theory aimed at applications, chemistry, specifically, appears to be its main focus (the first chapter centers around the C3v group in the examples).

Since it's a Dover book, you can get it on Amazon for only $14 (free two-day shipping with Amazon Student or free super saver on$35+ orders) and you can even preview it on Google Books.
https://www.amazon.com/Symmetry-Int...e=UTF8&qid=1388628173&sr=8-1&keywords=mcweeny