Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
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.
Lecture notes:
Monographs:
in general:
John S. Rose: A Course on Group Theory, Cambridge University Press (1978) [ISBN:9780521291422, pdf]
Marshall Hall, The Theory of Groups, Macmillan (1959), AMS Chelsea (1976), Dover (2018) [ISBN:978-0-8218-1967-8, ISBN:9780486816906]
Israel Grossman, Wilhelm Magnus: Groups and Their Graphs, Random House (1964), reprinted by: Anneli Lax New Mathematical Library 14 MMA (1992) [jstor:10.4169/j.ctt19b9mc7]
Joseph J. Rotman, An Introduction to the Theory of Groups, Springer (1995) [doi:10.1007/978-1-4612-4176-8, pdf]
in a more general context of algebra:
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]
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, 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, 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]
Wu Ki-Tung: Group Theory in Physics, World Scientific (1985) [doi:10.1142/0097]
Shlomo Sternberg: Group Theory and Physics, Cambridge University Press (1994) [ISBN:9780521558853]
J. F. Cornwell: Group Theory in Physics – An Introduction, Academic Press (1997) [ISBN:9780121898007, doi:10.1016/B978-0-12-189800-7.X5000-6]
A. P. Balachandran, S. G. Jo, Giuseppe Marmo: Group Theory and Hopf Algebras – Lectures for Physicists, World Scientific (2010) [doi:10.1142/7872]
See also:
Formalization in univalent foundations of mathematics (homotopy type theory with the univalence axiom)
and implementation in Agda:
On aspects of group theory seen inside homotopy theory/-group theory:
Last revised on April 13, 2025 at 09:19:32. See the history of this page for a list of all contributions to it.