I have a good understanding of linear algebra, so I picked up the schaum's outlines on group theory, as it is my primary source of learning. I don't quite see the big picture, what exactly is a group? can anyone give examples, thx.
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examples:
1. the integers under usual addition
2. the rationals under addition
3. Q*, the nonzero rationals under usual multiplication
4. positive real numbers under multiplication
5. nth roots of unity (the other 4 are infinite groups; this one is finite)
nonexamples:
1. integers under usual multiplication (not every integer has an integer inverse)
2. the set of nonzero real numbers, where the operation is given by a*b = ba^2
If you've picked up any text on group theory, you should know what it is. Formally, it is a set S together with an operation * on the set which satisfies 3 (or 4) basic axioms:
Associativity: For all a, b, and c in S, a*(b*c) = (a*b)*c
Identity: There exists unique e in S such that for all a in S e*a = a*e = a
Inverses: For all a in S, there exists a unique b in S such that a*b = b*a = e
By calling * an operation on the set, it is sometimes assumed implicitly that the axiom of closure holds
Closure: For all a, b in S, a*b is in S
I can't remember off the top of my head, but some of the wording in the axioms above is superfluous. You can certainly prove that the identity element is unique without requiring it in the axiom. Suppose e and f are two elements such that for all a:
a*e = a*f = e*a = f*a = a
well you know that e = e*f = f, so the identity is obviously unique. I can't remember if you can prove that if e*a = a, then a*e = a without assuming that the identity element is both a left and right identity, but you probably can. You might be able to prove similar things for inverses, i.e. you might be able to prove that inverses are unique without including it in your axioms, and you might be able to prove that the left inverse is the right inverse without including it in your axiom. Anyways, the above conditions do form necessary and sufficient conditions for a pair (S, *) to be a group.
Normally, you won't think of a group as a pair of a set and an operation, you'll just think of it as the set. Or you could always define things like inclusion: x is an element of G = (S, *) if and only if x is in S.
Again, if you have a text on groups, you should know some examples. The group Z_{n} consists of the set {0, 1, ..., n-1} and the group operation is addition modulo n. Another example is the permutation group S_{X} which consists of the set of all bijections from the set X to the set X, and the group operation is composition of functions. Suppose you have a tetrahedron. Then there are certain ways you can rotate it so that the tetrahedron ends up in the same place. These rotations form a set, and together with the operation of composition (composing one rotation with another means doing one followed by another) they form a group. The set of invertible (n x n)-matrices form a group with normal matrix multiplication.
One application of group theory is for studying sets of invertible functions that can be composed with each other.
For example, if you were doing Euclidean geometry, you might like to study isometries of the plane (combinations of translations, rotations, and reflections).
Then, if you draw a really important square, you might want to consider the subgroup of isometries that preserve that square -- i.e. the symmetry group of that square.
Then, you want to draw some new shape that shares the symmetry of your square. So, you draw just part of your new shape, and take its orbit under the action of the symmetry group to get your new shape!
Or, maybe the square isn't really important, you just want to watch how the isometries push it around the plane. Since different isometries might do the same thing to the square (e.g. the symmetries of the square do the same thing as the identity transformation), you would want to look at the quotient group of the isometries of the plane modulo the symmetries of the square.
YOU SAY YOU KNOW LINEAR ALGEBRA SO YOU KNOW ABOUT MATRICES AND THEIR RULE OF MULTIPLICATIONS AND YOU KNOW NOT ALL MATRICES HAVE INVERSE FOR MULTIPLICATION.
IN oops, a group, all elements do have inverses. so an example of a group is all those n by n matrices which do have multiplicative inverses.
or as matt said, those n by n matrices, say that preserve length, are certain invertible transformations of n space.
or for a finite example take all isometries of a cube, or an icosahedron, where the operation is composing isometries.
Hi, I'm not here to tell you what a group is, but to point to you the existence of this free e-book on abstract algebra that I am currently using to learn about groups, and am quite satisfied with.
Groups are possibly the nicest algebraic object to work with. Algebraic means simply (in this post) "not analysis, nothing to do with limits". They combine a small set of axioms, so they are easy to deal with, with a strong set of axioms (at least if we think about finite groups. Let us classify groups as either finite or infinite. Hopefully you are aware that a group is primarily a set of elements, and the finite/infinite referes to whether this underlying set is finite or infinite. Now, you are familiar with several groups, and they are of the infinite variety. I don't particularly like infinite groups in elementary courses, but it's good to be at least aware of them. Them being the integers, rational, real of complex numbers under addition, or the nonzero ones under multiplication. As you know linear algebra there are also the set of nxm matrices under addition, or the n by n invertible matrices under matrix mulitplication. These (especially the matrix groups) are very important, and I am not dismissing them when i say that i don't like them in an elementary course. My problem with them is that in order to talk about any of them and to prove anything about them at all we often need very complicated and highly specialized results. Indeed it was a high point of algebra to even find and classify the "representations" of a very small subset of the possible matrix groups (the Lie groups). Anyway, enough of that. Let me get back to FINITE groups since all finite groups share some incredible properties, so from now on all groups will be FINITE.
Any group is a subset of the permutaions of some set of objects (the cayley hamilton theorem) and it behoves us to study these so called permutation groups since they are esay to think of and contain all the information about all possible groups (remember we only care about finite ones). Thus we declare the following:
let X be a finite set. We will use Bij(X) to mean the set of all bijections (rearrangements) of X. Recall, if necessary, that a bijection of X is a function from X to X that is both injective (one to one) and surjective (onto). Note that since X is finite f: X --> X is bijective iff it is injective iff it is surjective.
Bij(X) is the canonical example of a group. The composition is composition of functions and if X has n elements then Bij(X) is also called S_n.
So far I've not explained what a group is, let us correct that.
Let X and Bij(X) be as above. G is a "a concrete group acting on G" iff
G is a subset of Bij(X).
G contains the identity map of X, ie the map i satisfying i(x)=x for all x in X
if f and g are in G so is fg the comopsition (remember these are just functions)
if f is in G so is the inverse function.
I hope you understand all those ideas from the ideas of functions in general
if it helps, think of these as shuffles of a deck of cards.
here is some more of this with some pitcures and everything:
www.maths.bris.ac.uk/~maxmg/teaching.html[/URL]
has some notes on it.
as you know abuot matrices, then we can talk about "representations" of groups, whihc we can think of as ways of "realizing" a group. often the confusion with groups the first time round is that they are too abstract, which is why i emphasize the concrete approach above, and then twe can do abstract later. matrices are another way of making concrete these abstract objects. which way would you like to take it?
in the absence of an immediate reply let me give some examples of groups as concrete groups.
the simplest example of a group is the so callled trivial group that just contains one element, the identity map. this is a subgroup of any group, and the X it acts on can be completely arbitrary.
next up are cyclic groups: imagine a set of n playing cards. We start with the simplest shuffle: let t be the shuffle that takes the top card and puts it on the bottom. this "generates a group". Namely we take t and try to find the smallest group that contains it. what are the rules of a group? we need to have the identity so we have
{i,t}
so far. what next? well, we need the closure under composition, in particluar that if we do t then t again we get something in the group. Let t^2 denote doing t twice. of course we need to denote doing t r times (where r is some integer) one after the other so t^r will do nicely, thus weh potentially have
{i,t,t^2,t^3,....}
an infinite set. but what happens if we do t n times where n was the number of the cards in the deck? we get the identity map, and hopefully you see that this is the first time that composing t with itself gives the identity map. so G now contains
{i,t,t^2,..,t^{n-1}}
n elements, and it is all closed under composition since t^r composed with t^s is t^{r+s} where we know we can knock of multiples of n from r+s.
this just leaves inverses. but notice that if we compose t^r with t^{n-r} we get the identity so actually we have all the inverses too. so it is a group.
if you like you can also think of this as the rotation group of a regular n sided polygon with t being rotation by 2pi/n radians.
or if you want to think as an abstract group this is also the set of complex n'th roots of unity with t being exp{2pi/n}
if you have been stidying groups you may have noticed that i have omitted to mention anything about associativty. this is because i get associativty for free since i am talking purely about composing functions and that is an associative process. this is another good reason to think of groups as permutations of a set.
to show that we really have lost nothing in thniking like this.
Let G be a concrete group acting on a set X, then G is a group in the abstract sense simply by forgetting the X.
Now let H be an abstract group then we can make it a concrete group as follows:
let X be the underlying set of elements of H, then H acts by permuting the elements of X.: let h be in H and define a permutation f_h of X by f_h(x)=hx for all x in X. this is a bijection since hx=hy implies, after cancelling the h on the left that x=y so it is injective and on a finite set this means it is a bijection.
so we can pass from abstract to concrete groups at will. notice that if we do this by taking a concrete group and forgetting the set it acts on then when we "concretify" it again the set it now acts on isn't the same set as that which it originally acted on. thus the same abstract group may act in many different yet essentially equivalent ways. but this is n't a surprise since i already showed that the cyclic group above could be realized as permutations of a deck of cards or as the rotations of an n-gon.
the link i gave was to a book assuming only matrices and determinants, and aimed at upper level undergrads in sophistication, or mathematically naive grad students.
my notes were aimed at someone who has done a first basic short course in groups and is finding it all a little too abstract. say the kind of person who, despite being told otherwise, still thinks that ab=ba for all elements of a group. there is a larger set of notes that starts from scratch but i can't get access to them at the minute. i know no knot theory and i don't think my concatenation of strings was written with that in mind. there are certainly links to "links" and braids and things physicists care about but i don't understand them and think that such discussions, even if i were to know anything about them, would be unsuitable for that set of notes.
what do you mean by 'practical'? it's maths, maths is about proving things. a "practical" question might well n eed to assume you are familiar with chemistry or physics or electronics. which we cannot do since it is supposed to be an introduction for mathematicians.
if you define what you mean by practical then perhaps someone can give a problem to you. what, for instance, is impractical in determining the symmetry group of the tetrahedron? or showing that the symmetry gourp of the cube is the same as the octahedron? or finding some conjugacy classes of a group? what was not practical about working out D_3 as matrices? i don't recall if all of these were included.
if that's the case, i believe my understanding about group is far too little to bring me to some insights on the use of group theory. i need to learn more to know what problems need group theory to solve...
group theory came about from the wrok on galois theory, but if you're new to groups that is a little too advanced. some elementary uses of groups are in coding and cryptography. as always maths requires a little patience so that you dont try to run before you walk.
how about this:
let p be a prime. consider the integers mod p, that is the set of numbers
0,1,2,..,p-1 with addition and multiplication defined modulo p (ie after adding or multiplying knock off multiples of p 'til you're back in that set, eg 3+(p-1)=2, or (p-1)(p-1)=p^2-2p+1=1)
show that this is a group under addition. show that the nonzero elements are a group under multiplication (the only hard thing is inverses but if you know a little number thoery it's straight forward). deduce fermat's little theorem: a^p=a mod p. there is exactly one element of multiplicative order 2, that is exactly one number x (not equal to 1) such that x^2=1. why? what is it? now try to prove wilson's theorem that (p-1)!=-1 mod p nb -1 is the same as p-1, and in general -r and p-r are the same element.
if you can, look up some cryptography to see how you can use this to create a simple private key cipher.
i'm not sure that the problems in other areas of science that use group theory are necessarily difficult, i think they are just unkown to mathematicians who tend to study things for their own sake. certainly people who are interested in chemistry or crystallography are interested in simple group theory problems, though many require some knowledge of representations of groups. a representation (of an abstract group) is when the group acts on a vector space as a group of matrices.. this for instance can tell you about the normal modes of vibration of the atoms in a molecule, though i don't konw how.
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