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Motivation

I was taking a course in abstract algebra. The professor is explaining that we need abstract algebra to convert difficult problems into algebraic way of solving. That's nice, but he started explaining groups and then he said we can make up our own number systems. I didn't quite understand the essence of creating a new number systems.

Question

Could anyone give an example that a problem that can be solved by creating a new group?

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Pawan Kumar
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    Possible duplicate of Are there real world applications of finite group theory? – José Carlos Santos Jan 14 '18 at 09:13
  • they used groups structures in chemistry and biology to find symmetry of all positions of atom to make drugs and antibiotics, etc...to find cancer and illness.. – DeepSea Jan 14 '18 at 09:15
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    For something to only be solvable by inventing new groups is probably difficult. Is it ok with an answer which gives example where a problem becomes easier to solve with group theory? – mathreadler Jan 14 '18 at 09:16
  • @mathreadler, Sir that would be a great help. Actually my Prof. in Complex number analysis explained that by introducing "i" in our number system we can solve any polynomial equation easily and also understanding the Complex plane with polar representation made me think solving problems by visualizing geometry behind several operations and functions. Now suddenly I cant wrap around this new Number systems concept. – Pawan Kumar Jan 14 '18 at 09:31
  • This is only meant to be a partial answer, but as an elementary example of why you want to consider 'other number systems' even to solve the equation 3x=2 (mod 5) in number theory you are already working over a different 'number system' (i.e. the integers modulo 5). – loch Jan 14 '18 at 09:50
  • There is a "magic trick" such that to understand how it works, one lust understand the chinese remainder theorem for groups $\mathbb{Z}/ab\mathbb{Z}$. I don't know if that counts though – Maxime Ramzi Jan 14 '18 at 10:09

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In algebraic topology groups are used to study topological spaces... But this is more advanced. ..

For example, the spheres $S ^n $ and $S ^m$ aren't homeomorphic ( topologically the same) for $m\not =n $, because they have different homology groups. ..

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The Pell equation is the equation $x^2-dy^2 = 1$ where $d$ is a positive non-square integer. The set of all solutions of the Pell equation is infinite. The first solution $(x_1, y_1)$ of the Pell equation is called the fundamental solution.

One may rewrite the Pell equation as $x^2-dy^2=(x+y\sqrt d)(x-y\sqrt d)=1$

so that finding a solution comes down to finding a nontrivial unit in the ring $\mathbb Z[\sqrt d]$. This reformulation implies that once one knows a solution, fundamental solution, of the Pell equation, one can find infinitely many others. More precisely, if the solutions are ordered by magnitude, then the $ n th$ solution $(x_n, y_n)$ can be expressed in terms of the fundamental solution by

$x_n+y_n \sqrt d=(x_1+y_1\sqrt d)^n$ for $n >1$

Mustafa
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