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I have read calculus. Now I'm reading algebra. What is the application of groups, rings and modules in life?

When we study derivations we know several applications. What about groups, rings and modules? Thanks.

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closed as too broad by Alexander Gruber Jun 19 at 1:39

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs.If this question can be reworded to fit the rules in the help center, please edit the question.

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@mesel In a free group generated by an apple and a lemon. –  math137 Jun 18 at 19:07
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$\Bbb Z$, $\Bbb Q$ , $\Bbb R$ and $\Bbb C$ are rings, and most linear algebra classes study modules over $\Bbb R$ and $\Bbb C$. Applications of linear algebras and the integers abound everywhere... –  rschwieb Jun 18 at 19:35
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you may look at this to learn some generals about Mathematics of physical symmetry, which uses group theory. –  math137 Jun 18 at 19:40
    
Modules are the natural way to think about representations. Representation theory has lots of applications, especially in physics. –  spin Jun 18 at 19:57
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@72313 Here‌​'s an incomplete list of books. Here's a journal. –  Alexander Gruber Jun 19 at 7:07

2 Answers 2

Here is a real life example:

FIFA wants its association to last a long time. In their early history they thought about the ball being a cube. Notice that during that period of time only four colours of dye existed. Since obviously they couldn't repeat the same ball in two cups they where wondering how many world cups they could have using cube balls.

To do this they hired a mathematician who used this method and found out they could only get 240 world cups, since they realized this probably wouldn't be enough they decided to change the shape of the ball.

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If you are a music person maybe this would be relevant:

https://www.math.washington.edu/~morrow/336_09/papers/Ada.pdf

It is titled "The Framework of Music Theory as Represented with Groups"

So, in terms of an application of groups, someone versed in group theory could (potentially) jump into atonal music theory. I would regard this as an application of groups.

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