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phys documents (dragged) 77 - G which is also a group w.r.t...

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Chapter 13 Theory of groups 13.1 Introduction 13.1.1 DeFnition of a group G is a group for the operation if: 1. A,B G A B G : G is closed . 2. A,B,C G ( A B ) C = A ( B C ) : G obeys the associative law . 3. E G so that A G A E = E A = A : G has a unit element . 4. A G A - 1 G so that A A - 1 = E : Each element in G has an inverse . If also holds: 5. A,B G A B = B A the group is called Abelian or commutative . 13.1.2 The Cayley table Each element arises only once in each row and column of the Cayley- or multiplication table: because EA i = A - 1 k ( A k A i )= A i each A i appears once. There are h positions in each row and column when there are h elements in the group so each element appears only once. 13.1.3 Conjugated elements, subgroups and classes B is conjugate to A if X G such that B = XAX - 1 . Then A is also conjugate to B because B = ( X - 1 ) A ( X - 1 ) - 1 . If B and C are conjugate to A , B is also conjugate with C . A subgroup is a subset of
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Unformatted text preview: G which is also a group w.r.t. the same operation. A conjugacy class is the maximum collection of conjugated elements. Each group can be split up in conjugacy classes. Some theorems: • All classes are completely disjoint. • E is a class itself: for each other element in this class would hold: A = XEX-1 = E . • E is the only class which is also a subgroup because all other classes have no unit element. • In an Abelian group each element is a separate class. The physical interpretation of classes: elements of a group are usually symmetry operations which map a symmetrical object into itself. Elements of one class are then the same kind of operations. The opposite need not to be true....
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