TITLE: What are applications of rings & groups?
QUESTION [11 upvotes]: 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?

REPLY [5 votes]: 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.