Group Theory. PHYS Southern Illinois University. November 15, PHYS Southern Illinois University Group Theory November 15, / 7

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1 Group Theory PHYS Southern Illinois University November 15, 2016 PHYS Southern Illinois University Group Theory November 15, / 7

2 of a Mathematical Group A group G is a set of elements with a law of composition (multiplication) G G G such that 1 Associative: (gg )g = g(g g ) for all g, g, g G. 2 Identity Exists: There a unique element e G called the identity with the property that eg = g for all g G. 3 Inverse Exists: For every g G there exists an inverse element g 1 G such that gg 1 = e. A group G is called Abelian if gg = g g for all gg G. The group of finite if a finite number of elements are in G. Otherwise it is infinite. PHYS Southern Illinois University Group Theory November 15, / 7

3 of a Mathematical Group A group G is a set of elements with a law of composition (multiplication) G G G such that 1 Associative: (gg )g = g(g g ) for all g, g, g G. 2 Identity Exists: There a unique element e G called the identity with the property that eg = g for all g G. 3 Inverse Exists: For every g G there exists an inverse element g 1 G such that gg 1 = e. A group G is called Abelian if gg = g g for all gg G. The group of finite if a finite number of elements are in G. Otherwise it is infinite. Examples Z 2, Z, S N, GL(n), SL(n), O(n), SO(n), U(n), SU(n). PHYS Southern Illinois University Group Theory November 15, / 7

4 Group Isomorphisms +- Two groups G and G are said to be isomorphic (denoted by G = G ) if there is a invertible map φ : G G that preserves group multiplication: ab = c φ(a)φ(b) = φ(c) a, b, c G. Example SO(2) is isomorphic to U(1). PHYS Southern Illinois University Group Theory November 15, / 7

5 Order of Group A group element g G is of order n if it is the smallest integer for which g n = e. PHYS Southern Illinois University Group Theory November 15, / 7

6 Order of Group A group element g G is of order n if it is the smallest integer for which g n = e. A group G is periodic if every element is of finite order. PHYS Southern Illinois University Group Theory November 15, / 7

7 Order of Group A group element g G is of order n if it is the smallest integer for which g n = e. A group G is periodic if every element is of finite order. A group G is cyclic if there is an element a G such that every element g G can be written as g = a m for some integer m. PHYS Southern Illinois University Group Theory November 15, / 7

8 D 3 Dihedral Group The group D 3 is the set of six elements {e, a, b, c, d, d 1 } with group multiplication given by: a 2 = b 2 = c 2 = e ab = bc = ca = d ac = cb = ba = d 1. It is non-abelian, and the elements a, b, c are of order 2. PHYS Southern Illinois University Group Theory November 15, / 7

9 D 3 Dihedral Group The group D 3 is the set of six elements {e, a, b, c, d, d 1 } with group multiplication given by: a 2 = b 2 = c 2 = e ab = bc = ca = d ac = cb = ba = d 1. It is non-abelian, and the elements a, b, c are of order 2. Exercise (i) Show the following relationships: aba = c = bab, bcb = a = cbc, aca = b = cac. PHYS Southern Illinois University Group Theory November 15, / 7

10 D 3 Dihedral Group Exercise (ii) What is the order of d? and d 1? PHYS Southern Illinois University Group Theory November 15, / 7

11 D 3 Dihedral Group Exercise (ii) What is the order of d? and d 1? Exercise (iii) Show that the elements d and d 1 of D 3 can be identified with rotations in a plane through angles ±2π/3, while the elements a, b, and c can be identified with reflections through three axes making angles 2π/3 with each other. PHYS Southern Illinois University Group Theory November 15, / 7

12 D 3 Dihedral Group Exercise (ii) What is the order of d? and d 1? Exercise (iii) Show that the elements d and d 1 of D 3 can be identified with rotations in a plane through angles ±2π/3, while the elements a, b, and c can be identified with reflections through three axes making angles 2π/3 with each other. Exercise (iv) Finally, show that these elements contain all six permutations of the three vertices of an equilateral triangle. PHYS Southern Illinois University Group Theory November 15, / 7

13 Direct Product The direct product of two groups G 1 and G 2 is a new group denoted by G 1 G 2 which consists of all ordered pairs (g 1, g 2 ) with g 1 G 1 and g 2 G 2. Group multiplication is defined by (g 1, g 2 )(g 1, g 2) = (g 1 g 1, g 2 g 2). If G = G 1 G 2, then G is decomposed into the direct product of G 1 and G 2. PHYS Southern Illinois University Group Theory November 15, / 7

14 Direct Product The direct product of two groups G 1 and G 2 is a new group denoted by G 1 G 2 which consists of all ordered pairs (g 1, g 2 ) with g 1 G 1 and g 2 G 2. Group multiplication is defined by (g 1, g 2 )(g 1, g 2) = (g 1 g 1, g 2 g 2). If G = G 1 G 2, then G is decomposed into the direct product of G 1 and G 2. Example Z 2 Z 3 = Z6. PHYS Southern Illinois University Group Theory November 15, / 7

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