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advanced_tools:group_theory [2020/04/02 15:01]
130.246.243.49 Fix degree symbols
advanced_tools:group_theory [2020/09/07 05:18] (current)
14.161.7.200 [History]
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 -->​$(G,​\circ) = (\mathbb{Z},​+)$#​ -->​$(G,​\circ) = (\mathbb{Z},​+)$#​
  
-On of the simplest examples of a group is to take as the set $G$ the integer numbers $\mathbb{Z}$ . The group operations is then simply addition $\circ = +$.  ​+One of the simplest examples of a group is to take as the set $G$ the integer numbers $\mathbb{Z}$ . The group operations is then simply addition $\circ = +$.  ​
  
 The first check we have to perform is closure. ​ If we take two elements of $\mathbb{Z}$ The first check we have to perform is closure. ​ If we take two elements of $\mathbb{Z}$
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 <​blockquote>​His starting point was the theory of algebraic equations (such as the quadratic, or second-degree,​ equations that children learn in school). In the 1800s, French mathematician Évariste Galois discovered that, in general, equations of higher degree can be solved only partially. <​blockquote>​His starting point was the theory of algebraic equations (such as the quadratic, or second-degree,​ equations that children learn in school). In the 1800s, French mathematician Évariste Galois discovered that, in general, equations of higher degree can be solved only partially.
  
-But Galois also showed that solutions to such equations must be linked by symmetry. For example, the solutions to x5 = 1 are five points on a circle when plotted onto a graph comprised of real numbers along one axis and imaginary numbers on the other. He showed that even when such equations cannot be solved, he could still glean a great deal of information about the solutions from studying such symmetries.<​cite>​https://​www.nature.com/​articles/​d41586-018-03423-x</​cite></​blockquote>​+But Galois also showed that solutions to such equations must be linked by symmetry. For example, the solutions to $x^5 = 1are five points on a circle when plotted onto a graph comprised of real numbers along one axis and imaginary numbers on the other. He showed that even when such equations cannot be solved, he could still glean a great deal of information about the solutions from studying such symmetries.<​cite>​https://​www.nature.com/​articles/​d41586-018-03423-x</​cite></​blockquote>​
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advanced_tools/group_theory.1585832481.txt.gz · Last modified: 2020/04/02 13:01 (external edit) · Currently locked by: 172.69.23.112,216.244.66.248