1. MATH10001 Project 2 Groups part 1 http://www.maths.manchester.ac.uk/undergraduate/ ugstudies/units/2009-10/level1/MATH10001/

2. 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)

3. 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 ie.  is a way of combining two elements of a set together to get another element of the set.

4. 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.

5. 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 a n = a  a  …  a = e. (n times) If no such n exists we say that a has infinite order.

6. Examples 1. R \{0} with multiplication. 2. Z with addition. 3. {1, -1, i, -i} with multiplication. 4. Z n = {0, 1, 2, …, n-1} with modulo n addition. 5. G = set of symmetries of an equilateral triangle,  is ‘followed by’.

7. A B C lBlB lAlA lClC e = do nothing a = reflect in line l A b = reflect in line l B c = reflect in line l C r = rotate anticlockwise 120 o s = rotate anticlockwise 240 o A CB C BA B AC C AB B CA