GROUP THEORY
Poetically speaking,
“Group Theory is the branch of Mathematics that answers the question ‘ What is symmetry?’ “- Nathan C. Carter
Definition 1.1– A group ‘G’ is a set with two operations or functions, one called multiplication (or addition in some cases) m : G X G → G and the other called the inverse i : G → G. These operations should obey the following rules-
- Associativity : For every g, h, and k ∈ G, m(m(g,h),k) = m(g,m(h,k)).
- Identity : There is an element e in the group such that for every g ∈ G, m(g,e) = g and m(e,g) = g.
- Inverse: For every g ∈ G, m(g, i(g)) = e = m(i(g), g).
The inverse is denoted by i(g) = g−1.
The three rules above will then read as follows
1. (gh)k = g(hk).
2. ge = g = eg.
3. gg−1 = eg−1g.
Lemma 1– Let f : A → B, g : B → C, h: C → D be three functions. Then h ◦ (g ◦ f) = (h ◦ g) ◦ f.
Proof– Both sides are functions from A → D. To prove it we have to prove it true for any element a ∈ A.
(h ◦ (g ◦ f))(a) = h((g ◦ f)(a)) = h(g(f(a)))
Similarly,
((h ◦ g) ◦ f))(a) = (h ◦ g)(f(a)) = h(g(f(a))).
The set {I,R,R2, F1, F2, F3} is a group, where the multiplication rule is composition of symmetries.
Any symmetry, can be interpreted as a function R2 → R2, and composition of symmetries is just composition of functions. So this rule is associative.
Definition 1.2– The dihedral group Dn of order n is the group of symmetries of a regular n-gon.
With this notation, D3 is the group of the set of symmetries of an equilateral triangle.
ABELIAN GROUP
An Abelian group is a group in which the result of applying the group operation to any two group elements does not depend on the order in which they are written. So, this is also called a Commutative group.
The Abelian group axioms:-
Closure– For all a, b, in A, the result of the operation a.b is also in A.
Associativity– For all a, b, c in A, the equation (a.b).c = a.(b.c) holds.
Identity element– There exists an element e in A, such that for all elements a in A, the equation e.a = a.e holds.
Inverse element– For each a in A there exists an element such that a.b = b.a = e, where e is the identity element.
Commutativity– For all a, b in A, a.b = b.a.
| Convention | Operation | Identity | Powers | Inverse |
|---|---|---|---|---|
| Addition |  | 0 |  |  |
| Multiplication |  or  | 1 |  |  |
For the verification of a finite group being Abelian or not, a table (matrix), known as the Cayley table can be constructed in a similar fashion to a multiplication table. If the group is G = {g1 = e, g2,……, gn} under dot product, the (i, j) th entry of this table contains the dot product of gi and gj. The group is Abelian if and only if this table is symmetric about the main diagonal. If the group is Abelian then gi . gj = gj . gi.
Subgroups
Definition 2.1– Let G be a group and let H be a subset of G. We say that H is a subgroup of G, if the restriction to H of the rule of multiplication and inverse makes H into a group.
The restriction to H × H of m, the rule of multiplication of G, won’t even define a rule of multiplication on H itself, because there is no a priori reason for the product of two elements of H to be an element of H.
For example suppose that G is the set of integers under addition,
and H is the set of odd numbers. Then if you take two elements of
H and add them, then you never get an element of H, since you will
always get an even number.
Definition 2.2– Let G be a group and let S be subset of G. We say that S is closed under multiplication, if whenever a and b are in S, then the product of a and b is in S. We say that S is closed under taking inverses, if whenever a is in S, then the inverse of a is in S.
Proposition– Let H be a non-empty subset of G. Then H is a subgroup of G if and only if H is closed under multiplication and taking inverses. Furthermore, the identity element of H is the identity element of G and the inverse of an element of H is equal to the inverse element in G. If G is Abelian then so is H.
Proof- If H is a subgroup of G, then H is closed under multiplication and taking inverses by definition. So suppose that H is closed under multiplication and taking inverses. Then there is a well defined product on H. We check the axioms of a group for this product. Multiplication on H is associative, is to say that for all g, h and k ∈ H, we have (gh)k = g(hk). But g, h and k are elements of G and the associative rule holds in G. Hence equality holds above and multiplication is associative in H. As H is non empty we may pick a ∈ H. As H is closed under taking inverses, a−1 ∈ H. But then e = aa−1 ∈ H as H is closed under multiplication. So e ∈ H. Clearly e acts as an identity element in H as it is an identity element in G. Suppose that h ∈ H. Then h−1 ∈ H, as H is closed under taking inverses. But h−1 is clearly the inverse of h in H as it is the inverse in G. Finally if G is abelian then H is abelian.