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Group theory

In 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.

This article covers advanced notions. For basic topics, see Group (mathematics).

Various physical systems, such as crystals and the hydrogen atom, and three of the four known fundamental forces in the universe, 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[1] was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 2004, that culminated in a complete classification of finite simple groups.

If X is a set with no additional structure, a symmetry is a map from the set to itself, giving rise to permutation groups.

bijective

If the object X is a set of points in the plane with its structure or any other metric space, a symmetry is a bijection of the set to itself which preserves the distance between each pair of points (an isometry). The corresponding group is called isometry group of X.

metric

If instead are preserved, one speaks of conformal maps. Conformal maps give rise to Kleinian groups, for example.

angles

Symmetries are not restricted to geometrical objects, but include algebraic objects as well. For instance, the equation has the two solutions and . In this case, the group that exchanges the two roots is the belonging to the equation. Every polynomial equation in one variable has a Galois group, that is a certain permutation group on its roots.

Galois group

Given a structured object X of any sort, a symmetry is a mapping of the object onto itself which preserves the structure. This occurs in many cases, for example


The axioms of a group formalize the essential aspects of symmetry. Symmetries form a group: they are closed because if you take a symmetry of an object, and then apply another symmetry, the result will still be a symmetry. The identity keeping the object fixed is always a symmetry of an object. Existence of inverses is guaranteed by undoing the symmetry and the associativity comes from the fact that symmetries are functions on a space, and composition of functions is associative.


Frucht's theorem says that every group is the symmetry group of some graph. So every abstract group is actually the symmetries of some explicit object.


The saying of "preserving the structure" of an object can be made precise by working in a category. Maps preserving the structure are then the morphisms, and the symmetry group is the automorphism group of the object in question.

List of group theory topics

Examples of groups

(1991), Linear algebraic groups, Graduate Texts in Mathematics, vol. 126 (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-0941-6, ISBN 978-0-387-97370-8, MR 1102012

Borel, Armand

Carter, Nathan C. (2009), , Classroom Resource Materials Series, Mathematical Association of America, ISBN 978-0-88385-757-1, MR 2504193

Visual group theory

Cannon, John J. (1969), "Computers in group theory: A survey", Communications of the ACM, 12: 3–12, :10.1145/362835.362837, MR 0290613, S2CID 18226463

doi

Frucht, R. (1939), , Compositio Mathematica, 6: 239–50, ISSN 0010-437X, archived from the original on 2008-12-01

"Herstellung von Graphen mit vorgegebener abstrakter Gruppe"

; Stewart, Ian (2006), "Nonlinear dynamics of networks: the groupoid formalism", Bull. Amer. Math. Soc. (N.S.), 43 (3): 305–364, doi:10.1090/S0273-0979-06-01108-6, MR 2223010 Shows the advantage of generalising from group to groupoid.

Golubitsky, Martin

Judson, Thomas W. (1997), An introductory undergraduate text in the spirit of texts by Gallian or Herstein, covering groups, rings, integral domains, fields and Galois theory. Free downloadable PDF with open-source GFDL license.

Abstract Algebra: Theory and Applications

Kleiner, Israel (1986), "The evolution of group theory: a brief survey", , 59 (4): 195–215, doi:10.2307/2690312, ISSN 0025-570X, JSTOR 2690312, MR 0863090

Mathematics Magazine

La Harpe, Pierre de (2000), Topics in geometric group theory, , ISBN 978-0-226-31721-2

University of Chicago Press

(2005), The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry, Simon & Schuster, ISBN 0-7432-5820-7 Conveys the practical value of group theory by explaining how it points to symmetries in physics and other sciences.

Livio, M.

(1970), Abelian varieties, Oxford University Press, ISBN 978-0-19-560528-0, OCLC 138290

Mumford, David

2006. Symmetry and the Monster. Oxford University Press. ISBN 0-19-280722-6. For lay readers. Describes the quest to find the basic building blocks for finite groups.

Ronan M.

Rotman, Joseph (1994), An introduction to the theory of groups, New York: Springer-Verlag,  0-387-94285-8 A standard contemporary reference.

ISBN

; Lyndon, Roger C. (2001), Combinatorial group theory, Berlin, New York: Springer-Verlag, ISBN 978-3-540-41158-1

Schupp, Paul E.

Scott, W. R. (1987) [1964], Group Theory, New York: Dover,  0-486-65377-3 Inexpensive and fairly readable, but somewhat dated in emphasis, style, and notation.

ISBN

Shatz, Stephen S. (1972), Profinite groups, arithmetic, and geometry, , ISBN 978-0-691-08017-8, MR 0347778

Princeton University Press

(1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, ISBN 978-0-521-55987-4, MR 1269324, OCLC 36131259

Weibel, Charles A.

History of the abstract group concept

This presents a view of group theory as level one of a theory that extends in all dimensions, and has applications in homotopy theory and to higher dimensional nonabelian methods for local-to-global problems.

Higher dimensional group theory

This package brings together all the articles on group theory from Plus, the online mathematics magazine produced by the Millennium Mathematics Project at the University of Cambridge, exploring applications and recent breakthroughs, and giving explicit definitions and examples of groups.

Plus teacher and student package: Group Theory

(1911), "Groups, Theory of" , in Chisholm, Hugh (ed.), Encyclopædia Britannica, vol. 12 (11th ed.), Cambridge University Press, pp. 626–636 This is a detailed exposition of contemporaneous understanding of Group Theory by an early researcher in the field.

Burnside, William