advanced_tools:group_theory

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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] 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 $x^5 = 1$ 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> |

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advanced_tools/group_theory.txt · Last modified: 2020/09/07 05:18 by 14.161.7.200

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