Saturday, 18 May 2013

Group Theory

I'm currently reading S. Sternberg's “Group theory and physics”. I bought it about four years ago. I have notes written in it up to page 66, but only really understand about ¾ of pages 1-15 and 48-60. It's hard going, but the book does have the advantage that nearly everything important about Group Theory seems to be included.
But the basics are:
A mathematical Group consists of:
  • a Set of mathematical elements (numbers, matrices, rotations, etc.) that, in the abstract, we shall refer to here-on in by pronumerals (e, a, m ...) and
  • an operator (addition, multiplication, matrix multiplication, etc.) that we shall represent with the $\cdot$ symbol.
The Set of elements can be finite in size, or infinite. An example of this would be the (infinite Set of) Integers, and the addition operator. I will be showing an example of each Group property using this group.

But a random Set of elements and an operator can't be a Group unless it satisfies the following Rules or Laws:
  1. The elements must be closed under the operator. That means that if you for example, start with two Integers, and apply the Group operator addition, your answer will be another Integer. i.e. the answer lies in the Set of Group elements. If you get something that wasn't in your original Set, you either need to expand or contract your list of elements until everything obeys this rule.
  2. The operator must be associative. E.g. a(bc) = (ab)c for any elements a,b and c in the Group. (Remember that in the example below, $\cdot$ is Integer Addition.)
$1 \cdot (3 \cdot 4) = (1 \cdot 3 ) \cdot 4$
$1 + (3 + 4) = (1 + 3) + 4$
  1. There must be an identity element (hereby represented by e), that leaves other elements unchanged. $e \cdot a = a \cdot e = a$. In the Integer/Addition Group, this element is 0.
  2. For every element a in the group, there must be an inverse a-1 that “cancels out” the first element. $a a\cdot ^{-1} = e$. It turns out in Finite groups, there is only one inverse for each element.
$ 1 \cdot -1 = 0$
This doesn't seem useful at first glance, but it turns out that anything that obeys these rules, has certain behaviours. Once you discover something new about abstract groups using these rules, you automatically know that it will apply to any concrete example, be it addition of integers, multiplication of matrices, the symmetries of molecules, or even theories of the Universe.

The next post investigates how these rules can be applied to multiplication of Integers.

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