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Quantum Engineering Year 0: Mathematical Foundations Month 11Day 281

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Year 0·Month 11·Week 1

Day 281: Introduction to Groups — The Mathematics of Symmetry

Day 281 of 2,016~19 min read

Learning Objectives

•State and explain the four group axioms
•Verify whether a given set with operation forms a group
•Identify common groups: integers, rationals, matrices, roots of unity
•Construct and interpret Cayley tables
•Understand the connection between groups and physical symmetries
•Implement group operations computationally

Today's Schedule (7 hours)

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Overview Prerequisites Learning Objectives 1 The Group Axioms What is a Group Understanding Each Axiom Important Properties Abelian Groups 2 Examples of Groups Example 1 Integers Under Addition mathbbZ Example 2 Non-Zero Rationals Under Multiplication mathbbQ times Example 3 The General Linear Group GL_nmathbbR Example 4 Roots of Unity mu_n Example 5 Symmetries of an Equilateral Triangle The Dihedral Group D_3 Example 6 Integers Modulo n mathbbZ_n 3 Non-Examples Not Groups Non-Example 1 Natural Numbers mathbbN Non-Example 2 Integers Under Multiplication mathbbZ times Non-Example 3 Real Numbers Under Subtraction 4 Cayley Tables Cayley Table for mathbbZ_4 Properties of Cayley Tables Cayley Table for D_3 Non-Abelian 5 Quantum Mechanics Connection Symmetries as Groups Operators and Observables Commutators and Non-Commutativity Discrete Symmetries 6 Worked Examples Example 1 Verify mathbbZ_5 times is a Group Example 2 Show the Klein Four-Group Example 3 Matrix Group Verification 7 Computational Lab 8 Practice Problems Problem Set A Basic Verification Problem Set B Cayley Tables Problem Set C Challenging 9 Solutions to Selected Problems Solution A1 Solution C1 10 Summary Key Concepts Key Formulas Connection to Quantum Mechanics 11 Daily Checklist Understanding Skills Connections 12 Preview Day 282

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