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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