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group theory (changes) in nLab

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Context

Mathematics

Group Theory

group theory

Classical groups

Finite groups

Group schemes

Topological groups

Lie groups

Super-Lie groups

Higher groups

Cohomology and Extensions

Related concepts

quantum group

Idea

The study of groups.

There are many generalizations and related structures; for example the vertical categorifications of groups like 2-groups, horizontal categorifications as groupoids, groups with structure, like topological groups, Lie groups, thus also Lie groupoids, Lie infinity-groupoids; and noncommutative generalizations like quantum groups. Lie and algebraic group(oid)s have their infinitesimal precursors like formal groups, local Lie groups, tangent Lie algebras, tangent Lie algebroids etc. In the smooth context the relation between Lie groupoids and Lie algebroids is the subject of Lie theory.

Literature

Lecture notes notes:

James Milne, Group theory (2021) [[web](https://www.jmilne.org/math/CourseNotes/gt.html), pdf]

Textbook Monographs: accounts

in general general:

Joseph J. Rotman ,
Marshall Hall, The Theory of Groups, Macmillan (1959), AMS Chelsea (1976), Dover (2018) [[ISBN:978-0-8218-1967-8](https://bookstore.ams.org/view?ProductCode=CHEL/288), ISBN:9780486816906]
An Introduction to the Theory of Groups , Springer (1995) [[doi:10.1007/978-1-4612-4176-8](https://doi.org/10.1007/978-1-4612-4176-8), pdf]
Joseph J. Rotman, An Introduction to the Theory of Groups, Springer (1995) [[doi:10.1007/978-1-4612-4176-8](https://doi.org/10.1007/978-1-4612-4176-8), pdf]

in a more general context of algebra:

Anthony Knapp, Basic Algebra, Springer (2006) [[doi:10.1007/978-0-8176-4529-8](https://doi.org/10.1007/978-0-8176-4529-8), pdf]

and in relation to applications in (quantum) physics (cf. “Gruppenpest”):

Hermann Weyl, Quantenmechanik und Gruppentheorie, Zeitschrift für Physik 46 (1927) 1–46 [[doi:10.1007/BF02055756](https://doi.org/10.1007/BF02055756)]
Hermann Weyl, Gruppentheorie und Quantenmechanik, S. Hirzel, Leipzig, (1931), translated by H. P. Robertson: The Theory of Groups and Quantum Mechanics Dover (1950) [[ISBN:0486602699](https://store.doverpublications.com/0486602699.html), ark:/13960/t1kh1w36w]
Eugene P. Wigner: Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren, Springer (1931) [[doi:10.1007/978-3-663-02555-9](https://doi.org/10.1007/978-3-663-02555-9), pdf]
Eugene P. Wigner: Group theory: And its application to the quantum mechanics of atomic spectra, 5, Academic Press (1959) [[doi:978-0-12-750550-3](https://www.elsevier.com/books/group-theory/wigner/978-0-12-750550-3)]
Shlomo Sternberg, Group Theory and Physics, Cambridge University Press 1994 (ISBN:9780521558853)