Tuesday May 6, 2008 5:00 am Lethbridge Alberta

Mark Ronan. (2006). Symmetry and the Monster.

Chapter 4 Groups

• "In the mid-nineteeth centry the idea of a group was still quite new, and the first methods for finding 'simple' ones were to look at permutations." [p. 42]
  
  
• "... if one group is a sub-group of a larger group then the size of the smaller divides the size of the larger. This is called Lagrange's Theorem." [p. 43]
  
  
• "... if the divisor [of a group of size n] is a prime number then there is a subgroup having that size. For example, a group of 60 elements must contain sub-groups with two elements, three elements, and five elements, because 2, 3, and 5 are prime numbers that are divisors of 60." [p. 45]
  
  
• "... any group of symmetries can be treated as a group of permutations ..." [p. 49]
  
  
• "The group can be represented in many ways, ... Mathematicians frequently omit the distinction between an abstract group and some favorite way of representing it." [p. 51]
  
  
• "Groups can be seen as groups of symmetries, permutations, or motions, or can simply be studied and admired in their own right." [p. 51]

Chapter 5 Sophus Lie

• "Lie admired Galois's work enormously and wanted to do for differential equations what Galois had done for algebraic equations." [p. 57]
  
  
• "Lie's work on differential equations used multidimensional geometry. ... His use of geometry came about by treating the parameters of an equation as coordinates. [p. 59]
  
  
• "In Lie's case an equation could have many parameters, so he needed multidimensional space." [p. 62]
  
  
• "... a German high school teacher, Wilhelm Killing, ... was pursuing similar ideas on groups of transformations ... Killing started pursuing the idea of classifying Lie's groups of continuous transformations." [p. 65]
  
  
• "Killing had discovered a 'periodic table' of Lie groups. He placed them in seven families: A to G." [p. 66]

Chapter 6 Lie Groups and Physics

• " 'Lie theory is in the process of becoming the most important field in modern mathematics.' ... Quantum physics has made extensive use of Lie theory. ... Later in the book it will connect up with the Monster, via string theory, which is a way of combining quantum physics and general relativity theory." [p. 72]
  
  
• "The wave nature of an electron means ... It has to be treated as a wave encircling the nucleus, and that brings Lie groups into the picture. An atom exhibits spherical symmetry, and this suggests that the Lie group of rotations in three-dimensional space should be important in the structure of electron orbits." [p. 75]

Tags: mathematics, symmetry