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

Algebra

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

Lie groups
- Classical elliptic integrals, etc.
Harmonic analysis
Haar measure type arguments
Homogenous spaces

Combinatorics

burnside-cauchy-frobenius
transitive graphs
dense codes
analysis of block designs
finite geometry

Numerical analysis

efficient matrix multiplication

Number theory

galois cohomology

Topology

fundamental group, homotopy groups

Science

Statistics

dense block designs, analysis of block designs
examples of rapidly mixing markov chains

Biology

check biostats literature, algebraic statistics mostly uses abelian groups and commutative algebra, but probably some real groups too

Chemistry

Crystallography

Computer science

(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

Hrm, dunno

Material science

quasicrystals and texture recognition

Physics

Symmetry principles in general
Quantum groups, quantum mechanics
Heisenberg groups

Social science

Economics

i think game theory uses some group theory

Humanities

Art

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

Music

tons of musicology