An Introduction to Group Theory

Welcome STEEMathematicians, to my first post which is exclusively to do with maths! My field is Condensed Matter Physics, but I'm terribly fond of maths, so I do hope you trust in my writing. Right, lets get on with it, eh - today I'm going to introduce you to a beautiful concept in abstract algebra: the group. First, some motivation.

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The E8 Lie group, a perfectly symmetrical 248-dimensional object.

[ image credit ]

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For many simple, or idealised physical systems, it is possible to find symmetry operations which bring about some change of coordinates, while leaving the system itself invariant, or unchanged. Translations or rotations in space are characterised as continuous symmetries and give rise to conserved quantities, such as momentum or energy (remember conservation of momentum from physics?). Check out Noether's theorem (we will visit this in later posts). 

Another type of symmetry is a discrete symmetry, for which there are a finite number of operations. A good example of this type of symmetry is reflections in space, and we would expect that these give rise to discrete conserved quantities. 

Symmetry is a very powerful concept, both in physics and mathematics, as we can often exploit it to simplify problems. Now, since symmetry operations can be sequentially applied and reversed, they form what is known as a group, and are described within the mathematics of group theory.

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Definition of a group

A group G is a set of elements {X, Y, ...}, together with some rule of combination, that associates each ordered pair (X, Y) with a product, X · Y, for which the following axioms (the group axioms!) must hold true.

• Closure: For every pair of elements (X, Y) that belong to G, the product X · Y also belongs to G.
• Associativity: For all triples (X, Y, Z) belonging to G, it must be the case that X · (Y · Z) = (X · Y) · Z.
• Identity: There exists a unique element, I, belonging to G, for which it is the case that I · X = X · I = X.
• Inverse: For every element X of G, there exists an element X^-1 of G, such that X^-1 · X = X · X^-1 = I. 

A common notation is to write the elements of a group G as the set {G₁, G₂, ...}, or more briefly as {G_i}, a typical element being denoted by G_i. 

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Example: The Integers

A well known example of a group - and something we are all familiar with - is the integers under the group operation of addition. The identity, I is the integer 0, and the inverse of some integer X is just X^-1 = -X.

Try and prove that this is a group (i.e. it satisfies all the group axioms, given above). 

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Example: +1, -1 under multiplication

In this case, the identity is just 1, with the inverse of 1 being 1, and the inverse of -1 being -1. 

Convince yourself that this satisfies the group axioms!

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Number of elements in a group

The number of elements in a group can either be infinite, or finite. In the case where the group has a finite number of elements, we refer to it as a finite group (duh!). The number of elements it contains, is known as the order of the group, which we will denote by g. In the notation G = {G₁, G₂, ..., G_n}, the order of the group is g = n. 

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Commutation

Lets consider for a moment a group of rotations and reflections, with the identity as the null operation (i.e. do nothing). For this group, the operation (·) is taken to mean that the left-hand operation is carried out on the system after the right-hand one.

Thus, Z = X · Y means that the effect on the system by carrying out Z is the same as would be obtained by first carrying out Y and then carrying out X. The order of operations is important in this example. We use the standard notation to write operators acting on functions to the left of the functions.

We note that in the earlier example where we considered the set of all integers under addition, it is true that for all ordered pairs (X, Y) in G, we have Y · X = X · Y.

If any two particular elements of a group satisfy this, they are said to commute under the operation (·), and if all pairs of elements in a group satisfy this, then the group is said to be Abelian. The set of all integers under addition forms an infinite Abelian group. Neat. 

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Elementary results

Consider the expression X^-1 · (X · X^-1).

Using previous results, we are able to write X^-1 · (X · X^-1) = (X^-1 · X) · X^-1 = I · X^-1 = X^-1.

Now, we know that X^-1 belongs to G, and from the group axiom that demands X^-1 belongs to G (inverse exists), it is the case that there exists an element U in G, such that U · X^-1 = I.

We can write U · (X^-1 · (X · X^-1)) = U · X^-1 = I.

Transforming the left-hand side of this equation gives

U ·  (X^-1 · (X· X^-1)) = (U · X^-1) · (X · X^-1) = I · (X · X^-1) = X ·  X^-1,

which shows that X · X^-1 = I.

Similarly, we have X · I = X · (X^-1 · X) = (X · X^-1) · X = I · X = X.

It is also possible to demonstrate the uniqueness of the identity element I, and it can be shown that the inverse of any particular element is unique. Further elementary results that can be obtained by similar arguments are as follows.

Given any pair of elements (X, Y) belonging to G, there exist unique elements (U, V), also belonging to G, such that X · U = Y and V · X = Y. It must be the case that U = X^-1 · Y, and V = Y · X^-1, and they can be shown to be unique. This is referred to as the division axiom.

If X · Y = X · Z for some X belonging to G, then Y = Z. This is referred to as the cancellation law. 

Forming the product of each element of G with a fixed element X of G has the effect of permuting the elements of G. This is often written symbollically as G · X = G. If this were not the case, and X · Y and X · Z were not different (even though Y and Z were) then application of the cancellation law would lead to a logical contradiction. This result is called the permutation law. 

In any finite group of order g, any element X when combined with itself to form succesive elements, X² = X ·  X, X³ = X · X², ... will, after at most g₁ such combinations, produce the group identity I. Here X², X³, ... are elements of the group. If the actual number of combinations needed is m-1, i.e. X^m = I, then m is called the order of the element X in G. 

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Right, that's all for today folks. In the next post, we will spend some more time thinking about the permutation law, before moving on to consider mappings between groups - homomorphisms, isomorphisms, automorphisms, epimorphisms, all that good stuff. Cheers.