What is Group: Definition and 1000 Discussions

In mathematics, a group is a set equipped with an operation that combines any two elements to form a third element while being associative as well as having an identity element and inverse elements. These three conditions, called group axioms, hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The formulation of the axioms is, however, detached from the concrete nature of the group and its operation. This allows one to handle entities of very different mathematical origins in a flexible way, while retaining essential structural aspects of many objects in abstract algebra and beyond. The ubiquity of groups in numerous areas—both within and outside mathematics—makes them a central organizing principle of contemporary mathematics.Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups arise as symmetry groups in geometry but appear also in the Standard Model of particle physics. The Poincaré group is a Lie group consisting of the symmetries of spacetime in special relativity. Point groups describe symmetry in molecular chemistry.
The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s, who introduced the term of group (groupe, in French) for the symmetry group of the roots of an equation, now called a Galois group. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely, both from a point of view of representation theory (that is, through the representations of the group) and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory.

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  1. A

    Sub groups of the dihedral group

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  2. J

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  3. A

    A Quantum Gravity Research Group -- any standing in mainstream Physics?

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  4. tomdodd4598

    I Exploring Direct Sums of Lorentz Group Representations

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  5. Gerson J Ferreira

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  6. Ygggdrasil

    New group of eukaryotes discovered

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  7. T

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  8. Mr Davis 97

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  9. T

    I Images of elements in a group homomorphism

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  10. K

    I States and Group: Eigenvectors Represent One Dimension

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  11. K

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  12. Mr Davis 97

    I Why must the group N be finite in this result?

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  13. Mr Davis 97

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  14. Mr Davis 97

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  15. CharlieCW

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  16. Robin04

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  17. Mr Davis 97

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  18. Cryo

    A Nonlinear susceptibility and group reps

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  19. C

    MHB Proving matrix group under addition for associative axiom

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  20. Mr Davis 97

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  21. L

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  22. Y

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  23. Mr Davis 97

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  24. Mr Davis 97

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  25. karush

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  27. Mr Davis 97

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  28. Mr Davis 97

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  29. Mr Davis 97

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  30. Martin T

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  31. M

    MHB Which of the group axioms are satisfied?

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  32. M

    MHB Show that G is a subset of the symmetric group

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  33. C

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  35. Mr Davis 97

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  36. Mr Davis 97

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  37. T

    MHB What is the minimal dimension of a complex realising a group representation?

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  38. Mr Davis 97

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  39. K

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  40. N

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  41. M

    MHB Find the probability that each group has an equal amount of odd and even numbers

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  42. Alvan

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  43. S

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  44. J

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  45. S

    I What are the best resources for learning about Lorentz group representations?

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  46. J

    I Other ways to break the Higgs symmetry group

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  47. S

    I Understanding 4-Vector Representations in the Lorentz Group

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  48. Giulio Prisco

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  49. H

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  50. D

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