Group Theory | Brilliant Math & Science Wiki

Group theory is the study of groups. Groups are sets equipped with an operation (like multiplication, addition, or composition) that satisfies certain basic properties. As the building blocks of abstract algebra, groups are so general and fundamental that they arise in nearly every branch of mathematics and the sciences. For example:

Symmetry groups appear in the study of combinatorics overview and algebraic number theory, as well as physics and chemistry. For example, Burnside’s lemma can be used to count combinatorial objects associated with symmetry groups.

The molecule \( \ce{CCl_4} \) has tetrahedral shape; its symmetry group has 24 elements. Chemists use symmetry groups to classify molecules and predict many of their chemical properties.

Elliptic curve groups are studied in algebraic geometry and number theory, and are widely used in modern cryptography.

Points on an elliptic curve can be “added” using the rules above. The resulting group structure is the subject of much contemporary research. It can be used to classify solutions to the curve equation; also, the difficulty of certain computational problems related to the group makes it useful in cryptography.

Fundamental groups are used in topology, for instance, in knot theory, as invariants that help to decide when two knots are the same.

Every knot has an associated knot group. The knot groups of these three knots are different from each other, so none of these knots can be tangled or untangled into the others without cutting and pasting.