phys documents (dragged) 77

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

This note was uploaded on 01/30/2011 for the course PHYSICS 208 taught by Professor Ye during the Spring '10 term at Blinn College.

Ask a homework question - tutors are online