Groups are an abstract algebraic concept based on a set and a group law (a binary function which closes the set).
Group theory is the study of groups, structures which combine a set of elements and a group law which can combine two elements into a third. For the set and its law to be a group, the law must be associative, have a neutral element in the set, and each element must accept an inverse in the sert by the law.
Finite groups are an often used brick in the construction of asymmetric encryption, key exchange, and digital signature algorithms, e.g. ElGamal, Diffie-Hellman and DSA, respectively. Non-zero integers modulo a given prime, are a group, with multiplication as group law. Elliptic Curves are another kind of group, with its own specific law.
- Group theory on Wikipedia