# Real-world uses of Algebraic Structures

I am a Computer science student, and in discrete mathematics, I am learning about algebraic structures. In that I am having concepts like Group,semi-Groups etc...

Previously I studied Graphs. I can see a excellent real world application for that. I strongly believe in future I can use many of that in my Coding Algorithms related to Graphics.

Could someone tell me real-world application for algebraic structures too...

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Presumably your instructor can tell you. Have you tried asking that person first? –  KCd Sep 17 '11 at 17:45
@kCd My instructor told me that we can use that coding theory he doesn't explained it,I have to ask how it is tomorrow. –  EAGER_STUDENT Sep 19 '11 at 17:40
@KCd Is there something specific about this question that makes you think that the OP should ask an instructor first? Why doesn't the same logic apply to all questions here? What's the point of posting such a comment? –  Bill Dubuque Sep 19 '11 at 18:00
EAGER: Since your instructor mentioned coding theory and you mention that you're learning about groups, I suggest you look at Numbers, Groups, and Codes by Humphreys and Prest. The book A Concrete Introduction to Higher Algebra, by L. Childs, has sections which illustrate how different algebraic structures are used in devising codes, but it largely avoids group theory proper and may involve some algebraic structures (like finite fields) which you haven't seen yet in your course. –  KCd Sep 20 '11 at 1:19
Bill: See my previous comment. –  KCd Sep 20 '11 at 1:19

Here's one place to start. The Unreasonable Effectiveness of Number Theory contains the following interesting surveys. Their references should provide good entry points to related literature.

• M. R. Schroeder -- The unreasonable effectiveness of number theory in physics, communication, and music

• G. E. Andrews -- The reasonable and unreasonable effectiveness of number theory in statistical mechanics

• J. C. Lagarias -- Number theory and dynamical systems

• G. Marsaglia -- The mathematics of random number generators

• V. Pless -- Cyclotomy and cyclic codes

• M. D. McIlroy -- Number theory in computer graphics

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Finite groups are used in the analysis of molecular symmetry, and the wikipedia article on "molecular symmetry" is a reasonable starting point. Finite semigroups are connected to the theory of finite automata in computer science, but I am not aware of any source that treats the combination in an accessible way. (You will find a lot on finite automata.) Coding theory makes heavy use of finite fields and cyclic codes are based on finite rings. (My recommended source for this would be Pretzel's "Error-Correcting Codes and Finite Fields".)

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Group theory can be seen as at least one way to tackle the idea of symmetry.

For example, take something nice and symmetric like a circle (for the sake of argument, let's only consider rotational symmetries). What do you end up with?

First, you have 'actions' you can take on the circle which preserve the symmetry, for example rotating it by $\pi / 6$ radians. Second, you have the set of points of the circle itself, and third you have a way of combining them, IE a rotation of $\pi / 6$ with a starting point of $(1,0)$ gives you and ending point $(\frac{\sqrt{2}}{2},\frac{1}{2})$.

Now this is a mathematical example, but essentially any symmetry in our natural or constructed world will have something like this going on... this being called a 'group action'.

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The fact that electrons , positrons , quarks , neutrinos and other particles exist in the universe is due to the fact that the quantum state of these particles respects poincare invariance. Put in simpler terms, If Einstein's theory of relativity is to hold , Some arguments using group theory show that these kinds of particles that I mentioned respects Einstein's theory and that there's no fundamental reason they shouldn't exist. Scientists have used group theory to predict the existence of many particles .We use a special kind of groups called lie groups that are groups and manifolds in the same time.For example $GL(n,R)$ is a lie group of invertible linear transformation of the n-dimensional Euclidean space. Symmetry operations correspond to elements living inside groups. If you map these symmetry elements to the group of invertible (and Unitary) transformations of a Hilbert Space ( An infinite dimensional vector space where particle quantum state lives ) You can study how these particle states transforms under the action of the group

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