The mathematical field of abstract algebra known as group theory is important to many disciplines, because of the wide range of applications of group theory.
Mathematics
editAlgebra
edit- Galois theory of equations
- Steinberg and fundamental groups of rings
- Group rings as important examples in ring theory
- Easy counterexample for "noncomm domains have skew fields of fractions"
- "If G is torsion free, is kG a domain?" has generated tons of ring theory
- basically just check passman
- Maximal orders, and the Albert-Brauer-Hasse-Noether theorem more or less come down to crossed algebras, a simple application of groups to algebras
Analysis
edit- Lie groups
- Classical elliptic integrals, etc.
- Harmonic analysis
- Haar measure type arguments
- Homogenous spaces
Combinatorics
edit- burnside-cauchy-frobenius
- transitive graphs
- dense codes
- analysis of block designs
- finite geometry
Numerical analysis
edit- efficient matrix multiplication
Number theory
edit- galois cohomology
Topology
edit- fundamental group, homotopy groups
Science
editStatistics
edit- dense block designs, analysis of block designs
- examples of rapidly mixing markov chains
Biology
edit- check biostats literature, algebraic statistics mostly uses abelian groups and commutative algebra, but probably some real groups too
Chemistry
edit- Crystallography
Computer science
edit- (coding theory again)
- efficient network design
- Crypto
- Group theoretic analysis of block ciphers
- Counting arguments for stream ciphers
- Generalized Diffie Hellman problems (solve the word or conjugacy problem in some infinite nonabelian group)
Earth science
editHrm, dunno
Material science
edit- quasicrystals and texture recognition
Physics
edit- Symmetry principles in general
- Quantum groups, quantum mechanics
- Heisenberg groups
Social science
editEconomics
edit- i think game theory uses some group theory
Humanities
editArt
edit- symmetry based art, old pottery, fabrics, and modern escher style
Music
edit- tons of musicology