Download Groups part 1

MATH10001 Project 2
Groups part 1
http://www.maths.manchester.ac.uk/undergraduate/
ugstudies/units/2009-10/level1/MATH10001/
Group Theory
Groups have been a key part of mathematics for
nearly 200 years. They are central to the study of
algebra and symmetry and have wider applications
in Crystallography and Quantum Physics.
Evariste Galois (1811-1832)
When does the equation
ax = b
have a unique solution?
What do we mean by ax?
A Binary Operation  on a set A is such that
a, b  A  a  b  A
ie.  is a way of combining two elements of a set together
to get another element of the set.
Definition of a Group
A Group is a set G with a binary operation  that satisfies
(G1) Closure: for all a,b  G, a  b  G
(G2) Associativity: for all a,b,c  G, a  (b  c) = (a  b)  c
(G3) Identity: there exists an element e  G such that
e  a = a  e = a for all a  G.
(G4) Inverses: for every a  G, there exists an element a-1  G,
such that a  a-1= a-1  a = e.
Notice that a group doesn’t have to be commutative.
If the binary operation in a group G is commutative we say
that G is an abelian group.
If G is a finite set, the order of the group is the number of
elements in G, written as |G|.
The order of an element a is the smallest natural number n
such that
an = a  a  …  a = e.
(n times)
If no such n exists we say that a has infinite order.
Examples
1. R\{0} with multiplication.
2. Z with addition.
3. {1, -1, i, -i} with multiplication.
4. Zn = {0, 1, 2, …, n-1} with modulo n addition.
5. G = set of symmetries of an equilateral triangle,
 is ‘followed by’.
A
lC
e = do nothing
lB
C
B
lA
A
a = reflect in line lA
C
B
C
b = reflect in line lB
B
A
B
c = reflect in line lC
A
C
C
r = rotate anticlockwise 120o
A
B
B
s = rotate anticlockwise 240o
C
A