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.
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:
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:
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.
In our particular example, there does not exist a group of the first type of order 6; indeed, simply because a particular identity skeleton is conceivable does not in general mean that there exists a group that fits it.
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
A tetrahedron is the most simple of three space shapes since it consists of only four vertices (see figure below). The Greek tetra stands for four, and can still be found in some science words such as tetrachloride or tetravalent. The hedra is from the Greek for base, or seat. 
, or approximately 109.47°).