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Groups as symmetry

4,745 bytes added, 03:03, 1 March 2008
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In other words, symmetry is the fact that if you ''make some change'' (either in the object itself or in your perspective) the object looks exactly the same. The ''extent'' of symmetry can now be described by the number of such different perspectives you can use. For instance, an equilateral triangle possesses ''some'' symmetry: if you rotate by certain angles, it doesn't change. But rotating by an arbitrary angle does ''not'' send the equilateral triangle to itself. So the equilateral triangle isn't quite as symmetric as the circle.
Symmetries of an object is measured by the set of transformations that map the object to itself. Object can be replaced by a structure, or rule. We're used to thinking of symmetries of concrete objects (like mice, clocks, and historical monuments). But in physics, we're interested in the symmetry and invariance properties enjoyed by ''laws''is a deeper concept. In chemistry, we're interested in the symmetries of small things like molecules. 
===Symmetry from a fairness perspective===
This also relates to another important fact: equality, and fairness, translate in group theory to invariance under a certain group action. The group action is the one that permutes the elements.
===Symmetry from an indistinguishability perspective===
Another name for fairness is indistiguishability. You shouldn't be able to, ''a priori'', give any criterion that one thing satisfies, that the other doesn't.
''A priori'' distinguishability is much weaker than ''a posteriori'' distinguishability. The former is like saying: ''I cannot give beforehand a criterion that is satisfied by one and not by the other''. The latter is saying: ''Given the two, I cannot tell whether they are the same''.
For instance, if you believe that time has no natural origin, then any point in time looks like any other. You cannot give any ''a priori'' criterion that is satisfied by one point in time and not by the other. But given two points in time, you can certainly compare them: you can, for instance, say which one came earlier.
The fact that there's no ''a priori'' way of defining a point in time is related to the assumption that physical laws are invariant under time translation.
In a similar way, you may say that there is no ''a priori'' mathematical way of defining a particular unit of length. In other words, there is no criterion that is satisfied by one unit of length, and not by the other. However, ''given'' two units of length, you can certainly ask which is bigger.
The inability to distinguish between various lengths is the ''dilation-invariance'' of mathematical (particularly geometric) theorems: the fact that the laws of geometry are invariant under rescaling lengths.
==Where groups come in==
We saw above that to measure the extent of a symmetry, we should specify two things:
* ''What'' thing we want to be invariant: It could be the region in space occupied by an object (like, a triangle or circle). It could be the effect of applying a law. It could be a differential equation.
* ''What'' transformations we are allowing
Given these two, we can define the ''group of symmetries'' as those transformations that ''preserve'' whatever we are trying to keep invariant.
Why is the group of symmetries a group? Firstly, if <math>T_1</math> leaves something invariant, and <math>T_2</math> leaves something invariant, then so does the composite <matH>T_1 \circ T_2</math>. In other words, composing two things that don't move it, doesn't move it. So we can define a multiplication on the group of symmetries by composition.
===The multiplication is associative===
This is the fact that function composition is associative. Another way of thinking of it is: if you have instructions (1) and (2) on the first page and instruction (3) on the second page, that has the same net effect as having instruction (1) on the first page and instruction (2) and (3) on the second page. (The instructions just tell us what transformations to do).
So in hindsight, ''associativity'' was really an appropriate condition to have a reasonable notion of a group ''acting'' on something.
===The identity map is in===
Certainly! The identity map preserves everything you could dream of. So, the identity map is actually inside, and it is the identity element of the group.
There may be a tendency to dismiss the identity element as unimportant or uninteresting. However, this is akin to thinking that vacuum is uninteresting in physics. The identity element is right in the middle of the group, and though it doesn't have much character of its own, it is needed for practically everything.
The existence of inverses is a debatable point, but the rough idea is that if your transformation actually ''was'' a symmetry, you should be able to undo it. Symmetries that aren't reversible aren't genuine symmetries. The ''reverse'' transformation is precisely the inverse element in the group.
There ''are'' certain kinds of situations where we want to relax the assumption about invertibility. Groups ''without inverses'' (i.e. sets with associative binary operation having an identity element) are termed [[monoid]]s. Monoids also ''act'', just like groups do, but the actions aren't often dubbed symmetries. You can't call a one-way street symmetric.
==Measuring symmetry by groups==
The bigger the symmetry group, the more the symmetry. However, the ''number of elements'' in the symmetry group isn't always the best measure of how symmetry-rich the structure is. Rather, various measures of complexity of the group are better.
For example the straight line has quite a lot of ''symmetry'', but I guess not too many people would marvel at its symmetry. But a beautiful lattice picture with a huge symmetry group is worth noticing.
There are two general rules:
* The more structural conditions we impose, i.e. the more things we demand remain invariant, the less the symmetry: If you settled for a less-than-perfect notion of ''invariance'', you'd get more symmetry elements.
* The larger the universe of transformations we look in, the more the symmetry
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