# What are applications of rings & groups?

I am following a course in basic algebra, and we have covered rings & groups in class, but I am having trouble visualising them. Are there applications of group &/or ring theory that can be more easily visualized than the abstract object? For instance, are there objects, or properties of objects, that behave as elements of a group in physics, chemistry, or other fields?

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Are the geometric groups of any interest? For example, what about the group of rotations of some polygon? – 000 Dec 8 '12 at 22:19
The theory of Group Rings has important connections to other fundamental areas, such as Number Theory, Topology, K-Theory, Representation Theory, Homological Algebra and of course to finite and infinite Group Theory and Ring Theory. Applications outside mathematics occur in Mathematical Physics (Crystallography) and within the last years also in Coding Theory and Cryptography. – Amzoti Dec 9 '12 at 0:20
All examples of groups are fine! – Émile Jetzer Dec 10 '12 at 1:54

Dihedral groups arise frequently in art and nature. Many of the decorative designs used on floor coverings, pottery, and buildings have one of the dihedral groups as a group of symmetry. Corporation logos are rich sources of dihedral symmetry. Chrysler’s logo has D5 as a symmetry group, and that of Mercedes-Benz has D3. The ubiquitous five-pointed star has symmetry group D5. The phylum Echinodermata contains many sea animals (such as starfish, sea cucumbers, feather stars, and sand dollars) that exhibit patterns with D5 symmetry. Chemists classify molecules according to their symmetry. Moreover, symmetry considerations are applied in orbital calculations, in determining energy levels of atoms and molecules, and in the study of molecular vibrations.

Source : Contemporary Abstract Algebra, Gallian, Chapter 2

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Thanks for your answer, I like it very much because of your inclusion of several fields. – Émile Jetzer Dec 10 '12 at 0:53
Glad to help! If you can get your hands on the text, there are a few more applications scattered throughout the chapter (and I'm sure later throughout the book, though I haven't gotten so far). – MSEoris Dec 10 '12 at 1:22

Group theory may be viewed roughly as a general study of symmetry. In chemistry this applies to crystals via the study of crystallographic groups, and in art via wallpaper groups. For an example in physics, the Lie symmetry groups of partial differential equations play fundamental roles, e.g, governing conservation laws and separation of variables. See for example Weyl's Symmetry and Budden's Fascination of Groups.

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This post hinges on whether or not you consider a polygon an abstract object. That is a highly debatable philosophical claim. As a result, I'm submitting this answer on good faith.

The most basic thing that comes to mind for me is the group of rotations of a polygon about its center point $P_k$. For instance,

$$(P_{3},r):=\{r(n): n \in \mathbb{Z}\},$$

the group of rotations of a triangle about its center where every element is a rotation of $n$ radians. We have a fundamental identity: $$r(0)=r(2\pi)=r\left(\frac{2\pi}{3}\right).$$ This is of interest because it directly relates to a triangle's three lines of symmetry. In fact, . . .

If we define $(P_k,r)$ to be the group of rotations of a $k$-gon about its center, we have the following identity for all values of $k$: $$r(0)=r(2\pi)=r\left(\frac{2\pi}{k}\right).$$

This is because every polygon has as many lines of symmetry as it has sides. As a result, you can inscribe a polygon in a circle and 'cut' the polygon $k$ ways via its lines of symmetry. This shows us that every rotation is the angle $\dfrac{2\pi}{k}$ radians because we cut the circle into $k$ equal parts.

For you to explore: Consider $(P_k,r,r')$ where $r'$ is the set of reflections across a line through the center. Is it a ring? If so, what properties arise?

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Thanks for the extra challenge at the end; also, the precision of your answer is appreciated. – Émile Jetzer Dec 10 '12 at 0:55
@ÉmileJetzer, I'm glad you enjoyed! You proposed a very creative question. :-) – 000 Dec 10 '12 at 10:46