- #1
battousai
- 86
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Outside of pure mathematics, where is group theory used?
What are the applications of group theory?
- Context: Undergrad
- Thread starter battousai
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Discussion Overview
The discussion revolves around the applications of group theory beyond pure mathematics, exploring its relevance in various fields such as physics, chemistry, cryptography, musicology, and art. Participants share examples and contexts where group theory is utilized, highlighting both theoretical and practical implications.
Discussion Character
- Exploratory, Technical explanation, Conceptual clarification, Debate/contested
Main Points Raised
- Some participants note that group theory is extensively used in Quantum Physics, with references to works by Michael Tinkham and Volker Heine.
- Group theory is mentioned as a tool in Theoretical Physics, particularly in General and Special Relativity and Elementary Particles.
- One participant points out the application of group theory in solving the Rubik's cube, suggesting a wealth of resources available online.
- Another participant highlights the use of group theory in chemistry for studying molecular symmetry, particularly in crystallography and spectroscopy.
- Group theory's role in cryptography, including techniques like RSA encryption, is also discussed.
- Some participants mention the relevance of cyclic groups in musicology, specifically in the 12-tone system.
- Group theory is noted to have applications in algebraic topology, which can influence the design of computer networks and communication systems.
- Artistic applications of group theory are mentioned, including its use in symmetric design elements in textiles, printing, and architecture.
- One participant emphasizes the ubiquity of group theory in everyday life, citing examples such as counting, shuffling cards, and measuring time.
Areas of Agreement / Disagreement
Participants generally agree on the wide-ranging applications of group theory, but there is no consensus on a singular focus or hierarchy of importance among the various fields mentioned.
Contextual Notes
Some claims about specific applications may depend on further definitions or assumptions about the context in which group theory is applied. The discussion does not resolve the depth or implications of these applications.
Who May Find This Useful
This discussion may be useful for students and professionals interested in the interdisciplinary applications of group theory across physics, chemistry, cryptography, musicology, and art.
- #2
HallsofIvy
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It is used extensively in Quantum Physics. For example, both Michael Tinkham and Volker Heine have written books titled "Group Theory and Quantum Physics".
Also group theory can be used to determine which polynomial equations can be solved in terms of radicals. (That was, in fact, the main reason for the development of group theory.) That is still mathematics but not what I would consider "pure" mathematics.
Also group theory can be used to determine which polynomial equations can be solved in terms of radicals. (That was, in fact, the main reason for the development of group theory.) That is still mathematics but not what I would consider "pure" mathematics.
- #3
Karlx
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- 0
Yes, indeed.
Group theory is extensively applied in Theoretical Physics:
General and Spacial Relativity
Elementary Particles
Quantum Physics
Group theory is extensively applied in Theoretical Physics:
General and Spacial Relativity
Elementary Particles
Quantum Physics
- #4
SteveL27
- 797
- 7
You use group theory to solve Rubik's cube! I was going to post a link but there are so many I'll just give the Google search and you can pick and choose. There are books on the subject and even college courses.
http://www.google.com/search?sourceid=chrome&ie=UTF-8&q=rubik's+cube+group+theory
http://www.google.com/search?sourceid=chrome&ie=UTF-8&q=rubik's+cube+group+theory
- #5
Deveno
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it is also used in chemistry to study molecular symmetry (many chemicals have chemical properties that derive partially or wholly from the types of symmetry they possess). this is especialy important in crystallography and spectroscopy.
group theory is an important part of cryptography (the study of codes), and plays a prominent role in understanding such techniques as RSA encryption.
cyclic groups play a role in musicology, and form a mathematical basis for the 12-tone system.
group theory is important in algebraic topology, which finds application, among other things, in design of computer networks and communication systems.
finally, there is group theory in symmetric artistic design elements (the so-called "frieze groups" and "ornament groups") used in textiles, printing, wall-papers, tiling (especially in arabic mosques), and such common-place items as kaleidoscopes (reflections form a group).
in short, groups are everywhere! if you've ever "counted by 2's or 3's" you were using intuitively a natural group property of the integers. the shuffling of a deck of cards, rotating a sphere in space, even our way of using a circle to measure time, these are all groups at work in our everyday lives.
group theory is an important part of cryptography (the study of codes), and plays a prominent role in understanding such techniques as RSA encryption.
cyclic groups play a role in musicology, and form a mathematical basis for the 12-tone system.
group theory is important in algebraic topology, which finds application, among other things, in design of computer networks and communication systems.
finally, there is group theory in symmetric artistic design elements (the so-called "frieze groups" and "ornament groups") used in textiles, printing, wall-papers, tiling (especially in arabic mosques), and such common-place items as kaleidoscopes (reflections form a group).
in short, groups are everywhere! if you've ever "counted by 2's or 3's" you were using intuitively a natural group property of the integers. the shuffling of a deck of cards, rotating a sphere in space, even our way of using a circle to measure time, these are all groups at work in our everyday lives.
- #6
battousai
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- 0
Thank you all. I'm learning the basics of it right now and I was just wondering where the hell it's used 
- #7
mathwonk
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group theory is the mathematics of symmetry, so just think about the places where symmetry occurs.
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