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[edit]

Algebra[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[edit]

Statistics[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[edit]

Hrm, dunno

Material science[edit]

quasicrystals and texture recognition

Physics[edit]

Symmetry principles in general
Quantum groups, quantum mechanics
Heisenberg groups

Social science[edit]

Economics[edit]

i think game theory uses some group theory

Humanities[edit]

Art[edit]

symmetry based art, old pottery, fabrics, and modern escher style

Music[edit]

tons of musicology