Evariste Galois died in a duel aged 20, but left behind enough ideas to keep mathematicians busy for centuries. These involved the theory of groups, mathematical constructs that can be used to quantify symmetry. Apart from its artistic appeal, symmetry is the essential ingredient for scientists who dream of a future theory of everything. Group theory is the glue which binds the 'everything' together."
... Tony Crilly
So begins the four page "Introduction" by Manchester UK Maths Historian Tony Crilly at the beginning of his four-page Chapter 38, Groups, in his book "50 Mathematical Ideas You Really Need to Know" We will return to the rest of his wonderful introduction momentarily, but first a few words about this weblog.
So far I have been posting in a stream-of-consciousness fashion, that is to say whatever topic interests me at any one time makes my weblog. However, as we dig deeper into "reality" we need to apply structure, so we will do so without getting carried away as so:
The subjects of my posts have recently begun to revolve and will continue to cycle through various fields that interest me, notably and in this order, for now:
- Mathematics
- Mathematical Physics/Applied Mathematics (same thing?)
- Physics
- Engineering
- Astronomy
- Lunar colonization
Today's lesson is on Group theory, which is vitally important in our real world which is best described by gauge theory, which grew from quantum field theory, which grew from quantum mechanics and special relativity. So let's begin.
Crilly continues:
Symmetry is all around us. Greek vases have it, snow crystals have it, buildings often have it and some letters of our alphabet have it. There are several sorts of symmetry: chief among them are mirror symmetry and rotational symmetry. We'll look at just two-dimensional symmetry for the purposes of this introduction - all our objects live on the flat surface of a two-dimensional plane.
MIRROR SYMMETRY
Can we set up a mirror so that an objects looks the same in front of the mirror as in the mirror? The word MUM has mirror symmetry, but HAM does not; MUM in front of the mirror is the same as MUM in the mirror while HAM becomes MAH. A 2-D tripod has mirror symmetry, but the triskelion (tripod with feet) does not. The triskelion as the object before the mirror is right-handed but its mirror image in what is called the image plane is left-handed.
 |
| Triskelion |
ROTATIONAL SYMMETRY
We can also ask if there is an axis perpendicular to the page so the object can be rotated in the page through an angle and be brought back to its original position. Both the tripod and triskelion have rotational symmetry. The triskelion, meaning "three legs", is an interesting shape. The right-handed version is a figure which appears as the symbol of the Isle of Man and also on the flag of Sicily.
If we rotate through 120 degrees and 240 degrees the rotated figure will coincide with itself; if you closed your eyes before rotating it you would see the same triskelion when you opened them again after rotation.
The curious thing about the three-legged figure is that no amount of rotation keeping in the plane will ever convert a right-handed triskelion into a left-handed one. Objects for which the image in the mirror is distinct from the object in front of the mirror are called chiral - they look similar but are not the same. The molecular structure of some chemical compounds may exist in some right-handed and left-handed forms in three dimensions and are examples of chiral objects. This is the case with the chemical compound limosene which in one form tastes like lemons and in the other case like oranges. The drug thalidomide in one form is an effective cure of morning sickness in pregnancy but in the other form has tragic consequences.
Steve here. That was only the first page and a half of Crilly's introduction. Next up in the rotation, probably next week at this time, will be Groups(3) - Measuring symmetry, being the continuation of Crilly. We will be introducing Cayley tables, which will give us the first opportunity to measure symmetry, based on the work of Arthur Cayley around 1854.
They're not hard, and will show the amazing result that the tripod, and not the triskelion, is the more complicated of the two.
If you're chomping at the bit and can't stand the suspense, or simply wish to read ahead, here's some voluntary non-required pre-homework for the elite of brain, being the entry on Cayley tables from the EVER-BORING and love-of-subject-killing textbooky encyclopedia that is mind-numbing Wikipedia:
A Cayley table, after the 19th century British mathematician Arthur Cayley, describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table. Many properties of a group — such as whether or not it is abelian, which elements are inverses of which elements, and the size and contents of the group's center — can be easily deduced by examining its Cayley table.
A simple example of a Cayley table is the one for the group {1, −1} under ordinary multiplication:
| × | 1 | −1 |
|---|---|---|
| 1 | 1 | −1 |
| −1 | −1 | 1 |
Contents |
History
Cayley tables were first presented in Cayley's 1854 paper, "On The Theory of Groups, as depending on the symbolic equation θ n = 1". In that paper they were referred to simply as tables, and were merely illustrative — they came to be known as Cayley tables later on, in honour of their creator.
Structure and layout
Because many Cayley tables describe groups that are not abelian, the product ab with respect to the group's binary operation is not guaranteed to be equal to the product ba for all a and b in the group. In order to avoid confusion, the convention is that the first factor (termed nearer factor by Cayley) in any row of the table is the same, and that the second factor (or further factor) in any column is the same, as in the following example:
| * | a | b | c |
|---|---|---|---|
| a | a2 | ab | ac |
| b | ba | b2 | bc |
| c | ca | cb | c2 |
Cayley originally set up his tables so that the identity element was first, obviating the need for the separate row and column headers featured in the example above. For example, they do not appear in the following table:
| a | b | c |
| b | c | a |
| c | a | b |
In this example, the cyclic group Z3, a is the identity element, and thus appears in the top left corner of the table. It is easy to see, for example, that b2 = c and that cb = a. Despite this, most modern texts — and this article — include the row and column headers for added clarity.
Properties and uses
Commutativity
The Cayley table tells us whether a group is abelian. Because the group operation of an abelian group is commutative, a group is abelian if and only if its Cayley table is symmetric along its diagonal axis. The cyclic group of order 3, above, and {1, −1} under ordinary multiplication, also above, are both examples of abelian groups, and inspection of the symmetry of their Cayley tables verifies this. In contrast, the smallest non-abelian group, the dihedral group of order 6, does not have a symmetric Cayley table.
Associativity
Because associativity is taken as an axiom when dealing with groups, it is often taken for granted when dealing with Cayley tables. However, Cayley tables can also be used to characterize the operation of a quasigroup, which does not assume associativity as an axiom (indeed, Cayley tables can be used to characterize the operation of any finite magma). Unfortunately, it is not generally possible to determine whether or not an operation is associative simply by glancing at its Cayley table, as is the case with commutativity. This is because associativity depends on a 3 term equation, (ab)c = a(bc), while the Cayley table shows 2-term products. However, Light's associativity test can determine associativity with less effort than brute force.
Permutations
Because the cancellation property holds for groups (and indeed even quasigroups), no row or column of a Cayley table may contain the same element twice. Thus each row and column of the table is a permutation of all the elements in the group. This greatly restricts which Cayley tables could conceivably define a valid group operation.
To see why a row or column cannot contain the same element more than once, let a, x, and y all be elements of a group, with x and y distinct. Then in the row representing the element a, the column corresponding to x contains the product ax, and similarly the column corresponding to y contains the product ay. If these two products were equal — that is to say, row a contained the same element twice, our hypothesis — then ax would equal ay. But because the cancellation law holds, we can conclude that if ax = ay, then x = y, a contradiction. Therefore, our hypothesis is incorrect, and a row cannot contain the same element twice.
Exactly the same argument suffices to prove the column case, and so we conclude that each row and column contains no element more than once. Because the group is finite, the pigeonhole principle guarantees that each element of the group will be represented in each row and in each column exactly once.
Thus, the Cayley table of a group is an example of a latin square.
Constructing Cayley tables
Because of the structure of groups, one can very often "fill in" Cayley tables that have missing elements, even without having a full characterization of the group operation in question. For example, because each row and column must contain every element in the group, if all elements are accounted for save one, and there is one blank spot, without knowing anything else about the group it is possible to conclude that the element unaccounted for must occupy the remaining blank space. It turns out that this and other observations about groups in general allow us to construct the Cayley tables of groups knowing very little about the group in question.
The "identity skeleton" of a finite group
Because in any group, even a non-abelian group, every element commutes with its own inverse, it follows that the distribution of identity elements on the Cayley table will be symmetric across the table's diagonal. Those that lie on the diagonal are their own inverse; those that do not have another, unique inverse.
Because the order of the rows and columns of a Cayley table is in fact arbitrary, it is convenient to order them in the following manner: beginning with the group's identity element, which is always its own inverse, list first all the elements that are their own inverse, followed by pairs of inverses listed adjacent to each other.
Then, for a finite group of a particular order, it is easy to characterize its "identity skeleton", so named because the identity elements on the Cayley table are clustered about the main diagonal — either they lie directly on it, or they are one removed from it.
It is relatively trivial to prove that groups with different identity skeletons cannot be isomorphic, though the converse is not true (for instance, the cyclic group C8 and the quaternion group Q are non-isomorphic but have the same identity skeleton).
Consider a six-element group with elements e, a, b, c, d, and f. By convention, e is the group's identity element. Because the identity element is always its own inverse, and inverses are unique, the fact that there are 6 elements in this group means that at least one element other than e must be its own inverse. So we have the following possible skeletons:
- all elements are their own inverses,
- all elements save d and f are their own inverses, each of these latter two being the other's inverse,
- a is its own inverse, b and c are inverses, and d and f are inverses.
It is noteworthy (and trivial to prove) that any group in which every element is its own inverse is abelian.
Filling in the identity skeleton
Once a particular identity skeleton has been decided on, it is possible to begin filling out the Cayley table. For example, take the identity skeleton of a group of order 6 of the second type outlined above:
| e | a | b | c | d | f | |
|---|---|---|---|---|---|---|
| e | e | |||||
| a | e | |||||
| b | e | |||||
| c | e | |||||
| d | e | |||||
| f | e |
Obviously, the e row and the e column can be filled out immediately. Once this has been done, it may be necessary (and it is necessary, in our case) to make an assumption, which may later lead to a contradiction — meaning simply that our initial assumption was false. We will assume that ab = c. Then:
| e | a | b | c | d | f | |
|---|---|---|---|---|---|---|
| e | e | a | b | c | d | f |
| a | a | e | c | |||
| b | b | e | ||||
| c | c | e | ||||
| d | d | e | ||||
| f | f | e |
Multiplying ab = c on the left by a gives b = ac. Multiplying on the right by c gives bc = a. Multiplying ab = c on the right by b gives a = cb. Multiplying bc = a on the left by b gives c = ba, and multiplying that on the right by a gives ca = b. After filling these products into the table, we find that the ad and af are still unaccounted for in the a row; as we know that each element of the group must appear in each row exactly once, and that only d and f are unaccounted for, we know that ad must equal d or f; but it cannot equal d, because if it did, that would imply that a equaled e, when we know them to be distinct. Thus we have ad = f and af = d.
Furthermore, since the inverse of d is f, multiplying ad = f on the right by f gives a = f2. Multiplying this on the left by d gives us da = f. Multiplying this on the right by a, we have d = fa.
Filling in all of these products, the Cayley table now looks like this:
| e | a | b | c | d | f | |
|---|---|---|---|---|---|---|
| e | e | a | b | c | d | f |
| a | a | e | c | b | f | d |
| b | b | c | e | a | ||
| c | c | b | a | e | ||
| d | d | f | e | |||
| f | f | d | e | a |
Because each row must have every element of the group represented exactly once, it is easy to see that the two blank spots in the b row must be occupied by d or f. However, if one examines the columns containing these two blank spots — the d and f columns — one finds that d and f have already been filled in on both, which means that regardless of how d and f are placed in row b, they will always violate the permutation rule. Because our algebraic deductions up until this point were sound, we can only conclude that our earlier, baseless assumption that ab = c was, in fact, false. Essentially, we guessed and we guessed incorrectly. We, have, however, learned something: ab ≠ c.
The only two remaining possibilities then are that ab = d or that ab = f; we would expect these two guesses to each have the same outcome, up to isomorphism, because d and f are inverses of each other and the letters that represent them are inherently arbitrary anyway. So without loss of generality, take ab = d. If we arrive at another contradiction, we must assume that no group of order 6 has the identity skeleton we started with, as we will have exhausted all possibilities.
Here is the new Cayley table:
| e | a | b | c | d | f | |
|---|---|---|---|---|---|---|
| e | e | a | b | c | d | f |
| a | a | e | d | |||
| b | b | e | ||||
| c | c | e | ||||
| d | d | e | ||||
| f | f | e |
Multiplying ab = d on the left by a, we have b = ad. Right multiplication by f gives bf = a, and left multiplication by b gives f = ba. Multiplying on the right by a we then have fa = b, and left multiplication by d then yields a = db. Filling in the Cayley table, we now have (new additions in red):
| e | a | b | c | d | f | |
|---|---|---|---|---|---|---|
| e | e | a | b | c | d | f |
| a | a | e | d | b | ||
| b | b | f | e | a | ||
| c | c | e | ||||
| d | d | a | e | |||
| f | f | b | e |
Since the a row is missing c and f and since af cannot equal f (or a would be equal to e, when we know them to be distinct), we can conclude that af = c. Left multiplication by a then yields f = ac, which we may multiply on the right by c to give us fc = a. Multiplying this on the left by d gives us c = da, which we can multiply on the right by a to obtain ca = d. Similarly, multiplying af = c on the right by d gives us a = cd. Updating the table, we have the following, with the most recent changes in blue:
| e | a | b | c | d | f | |
|---|---|---|---|---|---|---|
| e | e | a | b | c | d | f |
| a | a | e | d | f | b | c |
| b | b | f | e | a | ||
| c | c | d | e | a | ||
| d | d | c | a | e | ||
| f | f | b | a | e |
Since the b row is missing c and d, and since b c cannot equal c, it follows that b c = d, and therefore b d must equal c. Multiplying on the right by f this gives us b = cf, which we can further manipulate into cb = f by multiplying by c on the left. By similar logic we can deduce that c = fb and that dc = b. Filling these in, we have (with the latest additions in green):
| e | a | b | c | d | f | |
|---|---|---|---|---|---|---|
| e | e | a | b | c | d | f |
| a | a | e | d | f | b | c |
| b | b | f | e | d | c | a |
| c | c | d | f | e | a | b |
| d | d | c | a | b | e | |
| f | f | b | c | a | e |
Since the d row is missing only f, we know d2 = f, and thus f2 = d. As we have managed to fill in the whole table without obtaining a contradiction, we have found a group of order 6: inspection reveals it to be non-abelian. This group is in fact the smallest non-abelian group, the dihedral group D3:
| * | e | a | b | c | d | f |
|---|---|---|---|---|---|---|
| e | e | a | b | c | d | f |
| a | a | e | d | f | b | c |
| b | b | f | e | d | c | a |
| c | c | d | f | e | a | b |
| d | d | c | a | b | f | e |
| f | f | b | c | a | e | d |
Generalizations
The above properties depend on some axioms valid for groups. It is natural to consider Cayley tables for other algebraic structures, such as for semigroups, quasigroups, and magmas, but some of the properties above do not hold.
Groups(1) - Gross Introduction
8 comments:
This is a comment I just left which the blogger gods removed and then after took out the HTML links in an attempt to have it reposted.
Hi Steve,
I particularly like this piece as symmetry being the initial focus. That is as Crilly reminds symmetry is all around us and yet something many people ignore as not to question its significance.
Possibly the first to consider such seriously were the Chinese with their philosophy of Taoism or Daoism, with Dao having ineffable qualities that prevent it from being defined or expressed in words. Taoism’s main principle is that of Yin Yang whose symbol is an expression of opposing and yet complimentary symmetries, where one cannot exist or rather have purpose without the other. I’ve often thought if perhaps the Chinese had realized that a better language for such expression would have been mathematics you wouldn’t have had to dream about colonies on the moon or other planets as you would have been born on one already.
I myself introduced the concept of symmetry in my own blog some time back, yet there I used the circle as my example, as my emphasis was to show how it relates to conservation as well as to give light to the fact each are metrics of quality, rather than quantity, being an aspect of mathematics which many overlook when they consider how and why it is so powerful in capturing much of the nature of reality. Just as a last comment you and I both come from a time where a physically realized triskelion held deep meaning for us (the 45 record adapter centre piece) :-)
Best,
Phil
P.S. Oh by the way there is one error I found in this excellent introduction and that was with the Latin motto “Quocunque Jeceris Stabit” meaning “ wherever you throw it, it will stand”, found around the triskelion where the word for ”throw it” should be Jeceris rather the Jeceric. Then again who am I to point out such trivialities:-)
Oh no Phil, DO throw out such trivialities, and thanks for catching that. We're a bit shy about Latin here in the Colyer household, some latent prejudice about languaages that lack the word: "the"
However, that wasn't my page that clicking on "Triskelion" links to, that's some page I got when Google Image-ing triskelion and it just touched my fancy.
Well, Phil, whether you know it or not, you sir and others like you (like me!) are the reason I'm getting into Groups, and go as slow as need be, yet no slower than required so as not to be patronizing, and also not to bore. :-)
The reason THAT is important to be is because I think you a fantastic Philosopher, and we need MORE Philosophers in Mathematical Physics, not less, as answering the WHY? is the job of such as those of you. The Scientist asks HOW?, a very relevant question, but WHY? seems to elude us.
Question us, question us out the wazoo, always, as that's your job, see? But how can you question if you do not understand the question, since the strange notation of Math Phys is somewhat intimidating, and amateurs need not apply?
Also, Phil, pls don't come back that you're not a "Pro" on Philosophy. Balderdash! "Amateur" means "self-studied," NOT "poor" or "bad,", and worse you're a polite Canadian (redundant) so let's park the humility at the door before we even start, m'kay? :-)
To whit and back on track:
The Philosopher, not the Physicist, is best poised to answer the following question, the following WHY? question, which is perhaps the most important question of them all:
WHY U(1)xSU(2)xSU(3) ?
(U(1), SU(2), and SU(3) being Groups)
That's THE question Phil, THE question that asks why the electroweak and strong forces and therefore the universe are and is the way they and it are.
But if you don't know the Maths, you can't understand the question, and if you don't understand it, how can you answer it?
Hi Steve,
Thanks again for having an appreciation of my philosophical acumen. Also as I mentioned this particular topic in respect to Mathematics I find intriguing for the stated reasons. More so I will follow your progress with this new focus of yours and take in as much as I am able. As for you wondering why a particular symmetry grouping seems to represent the order of things I can’t say I hold the same interest as my principle concern. Now don’t take me wrong as me considering such a revelation would not serve to be a significant advance and yet from my own perspective an explanation for such alignments rests with something more fundamental, as to ask from where they themselves emerge as to have them as defining limits respective of reality.
From that side of things I was wondering if you have read anything of what I find to be the true present day science philosophers, such as David Z. Albert or Harvey Brown. That is from my way of looking at things being that the scientists themselves have for the most part forsaken philosophy and the solution is not so much that more philosophers come to understand science and math, yet more scientists and mathematicians include philosophy into their way of thinking and not be so overly confident in that pure objectivity to be able carry the day. For me what is obvious is that nature is purposeful and the question to be answered, as to have looked square in the face, is to ask if purposefulness is a prerequisite to reality(regardless if by chance or intent) or an exception to it. That is when it comes to what is reality, is it just one fine tuning of all possibly realizable states or is it the only state of possibilities which can form to become a reality.
“It has been often said, and certainly not without justification, that the man of science is a poor philosopher. Why, then, should it not be the right thing for the physicist to let the philosopher to the philosophizing? Such might indeed be the right thing at a time when the physicist believes he has at his disposal a rigid system of fundamental concepts and fundamental laws which are also well established that waves of doubt cannot reach them; but, it cannot be right at a time when the very foundations of physics itself become problematic as they are now. At a time like the present, when experience forces us to seek a newer and more solid foundation, the physicist cannot simply surrender to the philosopher the critical contemplation of the theoretical foundations; for, he himself knows best, and feels more surely where the shoe pinches. In looking for a new foundation he must make clear in his own mind just how far such concepts which he uses are justified, and are necessities.”
-Albert Einstein, “Physics and Reality”, Journal of the Franklin Institute [Volume.221, No. 3, March 3, 1936]
Best,
Phil
No, Phil, I hadn't heard of Albert or Brown, but thanks for turning me on to them. To date the only Philosopher-Scientist I was aware of was Tim Maudlin at Rutgers and his book on how can Special Relativity and Quantum Entaglement exist in the universe at the same time.
From Wiki:
David Z Albert, Ph.D., is Frederick E. Woodbridge Professor of Philosophy and Director of the M.A. Program in The Philosophical Foundations of Physics at Columbia University in New York. He received his B.S. in physics from Columbia College (1976) and his doctorate in theoretical physics from The Rockefeller University (1981) under Professor Nicola Khuri.[1] Afterwards he worked with Professor Yakir Aharonov of Tel Aviv University.
Prof. Albert has published two books (Quantum Mechanics and Experience (1992) and Time and Chance (2000) ) and numerous articles on quantum mechanics. His books are both praised and criticized for their informal, conversational style, but he is routinely credited by both fan and critic as having a talent for communicating difficult, highly abstract concepts in ways that are accessible to the lay science reader.
Appearance in What the Bleep Do We Know!?
Prof. Albert appeared in the controversial movie What the Bleep Do We Know!? (2004). He was disappointed with the final product, claiming that the film's producers used selective editing to make him appear to agree with claims that he completely disagreed with.
According to a Popular Science article, he is "outraged at the final product." The article states that Albert granted the filmmakers a near-four hour interview about quantum mechanics being unrelated to consciousness or spirituality. His interview was then edited and incorporated into the film in a way that misrepresented his views. In the article, Albert also expresses his feelings of gullibility after having been "taken" by the filmmakers. Although Albert is listed as a scientist taking part in the sequel to What the Bleep, called "Down the Rabbit Hole", this sequel is a "director's cut", composed of extra footage from the filming of the first movie.
The "Down the Rabbit Hole" version features David as the first subject in the interview portion of the film. In that interview he expresses his disagreement with the major thrust of the original "What the Bleep Do We Know!?"
I'll investigate Brown tomorrow. I am reminded of Schrodinger however, who wished to be a Philosopher, but upon realizing that Austria had an excess of Philosophers in its Academia, chose the much lower populated area of Physics instead. Good that he made that choice? I think so, but later on he returned to Philosophy, at least in his writings. Have you read them, and what were your thoughts; your overall impression?
Hi Steve,
That introduction to Prof. David Albert hardly does him justice, as he has taken fundamental issues in Physics down to their bare bones to have them examined by way of what we think we know to be only what we are certain we know. Then from this standpoint he points out how our own beliefs turn out in terms of the way they would be if that were true.
Thus I would of course highly recommend his book “Quantum Mechanics and Experience”, as his examination of the subject, although in great depth, is presented in such a way that anyone who can follow and understand logical reasoning with due effort can grasp it. However, I would argue that such a depth of understanding is had by very few, with the vast majority of physicists themselves being part of that group. If for some reason you don’t pick up the book I would highly recommend having a look at the Talking Head discussion he had a few years back with Sean Carrol.
Best,
Phil
Heya Phil,
Thanks ever so much, as always. You continue to turn me on to great thinkers I sure wish I'd known about at a much earlier age, so it's much appreciated.
It should surprise no one that you're my current best friend in my lifelong exploration of Mathematical Physics, partly because we think alike, but also because Andrew Thomas, my FIRST best friend in Math Phys, has decided to tell Science to take a 3-yr hike and time-out while he decides to explore the similar field of Moog Synthesizer Music as the artist Caspar von Biergaaarten.
Andrew SAVED me Phil, because just when I was about to tell Quantum Mechanics and therefore Physics to take a hike (for what is Physics without Quantum Mechanics?), I chanced upon Andrew's What Is Reality? website, which remains in my opinion the clearest, most succinct, and tightest explanation of QM anywhere.
I look forward to welcoming Andrew back into the scientific community someday, and I'm sure you do as well.
Which reminds me. Considering that this current horrible winter of ours will eventually end (theoretically), I look forward to visiting and spending a few days with you this spring or summer in Toronto. Granted, you'd be better visiting HERE, where I can show you Einstein's home in Princeton and the near-infinite glory-fest that is New York City, but if you're a homebody like me, then "travel" ain't your thing, and I more than understand.
Regarding your reply, I believe NOTHING with complete confidence about whatever I read at Wikipedia about anyone or any organization, that is to say I take anything I read there re people or organizations with a HUGE grain of salt, since anyone can write there, so I defer to your better judgment of this apparently fine man I hadn't heard of until you introduced him to me, thanks.
I'll definitely put that book on my bucket list due to nothing other than your say-so, but should I rank it higher than Zen and the art of Motorcycle Maintenance?
I have moved beyond this post by the way Phil to Groups(3), so let's continue the discussion there if you would.
Bad link alert! Andrew Thomas' What Is Reality? website, there.
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