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

Question

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?

Answer 1

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. ..

Answer 2

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 $n >1$