What is Group theory: Definition and 378 Discussions

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.
Various physical systems, such as crystals and the hydrogen atom, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography.
The early history of group theory dates from the 19th century. One of the most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 1980, that culminated in a complete classification of finite simple groups.

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  1. Gerson J Ferreira

    Solid State Group theory paper suggestions for my classes

    I teach group theory for physicists, and I like to teach it following some papers. In general my students work with condensed matter, so I discuss group theory following these papers: [1] Group Theory and Normal Modes, American Journal of Physics 36, 529 (1968) [2] Nonsymmorphic Symmetries and...
  2. A

    Show injectivity, surjectivity and kernel of groups

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  3. T

    I How to properly understand finite group theory

    I do have a fair amount of visual/geometric understanding of groups, but when I start solving problems I always wind up relying on my algebraic intuition, i.e. experience with forms of symbolic expression that arise from theorems, definitions, and brute symbolic manipulation. I even came up with...
  4. T

    I Images of elements in a group homomorphism

    Why does the image of elements in a homomorphism depend on the image of 1? Why not the other generators?
  5. A

    I Adjoint Representation Confusion

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  6. Mr Davis 97

    Proving Cauchy's Theorem in Group Theory

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  7. T

    ##\phi(R_{180})##, if ##\phi:D_n\to D_n## is an automorphism

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  8. L

    Group Theory: Finite Abelian Groups - An element of order

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  9. CharlieCW

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  10. Martin T

    I About Arnold's ODE Book Notation

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  11. C

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  12. N

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  13. JuanC97

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  14. diegzumillo

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  15. S

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  16. U

    Are these homomorphisms?

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  17. H

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  18. The Bill

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  19. A

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  20. A

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  21. M

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  22. DrHix

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  23. L

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  24. A

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  25. A

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  26. Mr Davis 97

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  27. J

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  28. D

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  29. Edward Vogel

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  30. M

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  31. Luca_Mantani

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  32. Mr Davis 97

    I Proving an exponent law in group theory

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  33. Konte

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  34. BubblesAreUs

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  35. L

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  36. Kara386

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  37. Heisenberg1993

    A Real parameters and imaginary generators

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  38. Matejxx1

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  39. S

    A Applying group theory to multivariate eqs

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  40. J

    A How is the invariant speed of light enocded in SL(2,C)?

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  41. munirah

    A How to Multiply SU(4)XSU(2) Matrices to Form a 8x8 Matrix?

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  42. V

    Courses Does this group theory course look useful to a physicist?

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  43. munirah

    Understanding the Parameters of SU(4) and SU(2)

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  44. J

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  45. AJPsi

    A Has This Mysterious Mathematical Group Been Identified in Physics Research?

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  46. mertcan

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  47. P

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  48. B

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  49. matqkks

    I Group Theory: Unlocking Real-World Solutions for First-Year Students

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  50. matqkks

    MHB Group Theory: A Powerful Tool for Real World Solutions

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