A group is a set together with an associative operation which admits an identity element and such that every element has an inverse. Throughout the article, we use to denote the identity element of a group. A C D F G H I L N O P Q R S T Basic definitions Subgroup. A subset of a group which remains a group when the operation is restricted to is called a subgroup of . Given a subset of . We denote by the smallest subgroup of containing . is called the subgroup of generated by . Normal subgroup. is a normal subgroup of if for all in and in , also belongs to . Both subgroups and normal subgroups of a given group form a complete lattice under inclusion of subsets; this property and some related results are described by the lattice theorem. Group homomorphism. These are functions that have the special property that for any elements and of . Kernel of a group homomorphism. It is the preimage of the identity in the codomain of a group homomorphism. Every normal subgroup is the kernel of a group homomorphism and vice versa. Group isomorphism. Group homomorphisms that have inverse functions. The inverse of an isomorphism, it turns out, must also be a homomorphism. Isomorphic groups. Two groups are isomorphic if there exists a group isomorphism mapping from one to the other. Isomorphic groups can be thought of as essentially the same, only with different labels on the individual elements. One of the fundamental problems of group theory is the classification of groups up to isomorphism. Direct product, direct sum, and semidirect product of groups. These are ways of combining groups to construct new groups; please refer to the corresponding links for explanation. Types of groups Finitely generated group. If there exists a finite set such that then is said to be finitely generated. If can be taken to have just one element, is a cyclic group of finite order, an infinite cyclic group, or possibly a group with just one element. Simple group. Simple groups are those groups having only and themselves as normal subgroups. The name is misleading because a simple group can in fact be very complex. An example is the monster group, whose order is about 1054. Every finite group is built up from simple groups via group extensions, so the study of finite simple groups