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As I asked in previous question, I am very curious about applying Group theory. Still I have doubts about how I can apply group theory. I know about formal definitions and I can able to solve and prove problems related to Group theory.

But when comes to applications, I don't know where to start.

I surfed the net, and I can get these links....

http://www.math.uconn.edu/~kconrad/math216/whygroups.html

http://en.wikiversity.org/wiki/Topic:Group_theory

http://ezinearticles.com/?Why-Study-Math?---Group-Theory-and-Subparticle-Physics&id=1456420

Those explanations are really good. But the real problem I face is, all applications are of theoretical explanations, without a solved example which a beginner like me can understand.

When I went to Wikipedia, I learned about the solution of the Rubik's cube in at most 20 steps posted in http://cube20.org/

What I can understand?

Turning a cube upside down, it will still take the same number of moves to solve.(Symmetrical property).

Where I need assistance?

An example of showing how this symmetrical property of group theory works here.

So, if someone could give an example of how group theory is applied (in this or some other instance) it will be useful to me....

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    By "implementing", to you mean **applying**?2011-09-19
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    @Arturo Magidin yes. Like solving by taking real time example and showing.2011-09-19
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    If you are interested in applications of groups in general, you might want to check out the web page of one Vladimir Shpilrain - he researches ways of applying decision problems in group theory (and other parts of algebra) to cryptography. Very interesting.2011-09-20
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    @Swlabr: Is Shpilrain's page about braid group cryptosystem? I once attended a seminar, where the idea was explained. I'm not into word problems at all myself, but I might want to refer a few colleagues to that page.2011-09-20
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    I think it is just non-abelian groups in general. I mean, he's got a cryptography paper about Thompson's group, as well as Braid stuff too. http://www.sci.ccny.cuny.edu/~shpil/res.html is the page. (He's also got a paper using Tropical stuff, so it isn't all groups).2011-09-21

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