Groups and normal subgroups
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A group is a set G equipped with an associative binary operation, with an identity element, in which every element has an inverse. Beginning in the 1860s, mathematicians such as Peter Ludwig Mejdell Sylow (1832–1918), Émile Mathieu (1835–1890), and Marie Ennemond Camille Jordan (1838–1922) became fascinated by groups with finitely many elements. Families of such groups were soon identified, including the group of congruence classes modulo n and the dihedral group of isometries that leave a regular n-gon invariant.
A subgroup H of a group G is a subset that satisfies the group axioms when the operation is restricted to its elements. Let x be an element of G. We write xH for the set of elements of the form xh, where h belongs to H (and similarly Hx is the set of elements hx, where h belongs to H); here the operation is written multiplicatively.
Évariste Galois had already identified subgroups that play a special role, known as normal (or invariant) subgroups: they are those for which xH = Hx for every element x of G. Their importance lies in allowing the structure of a group to be reduced to the study of "smaller" groups, through what are known as quotient groups.
Thus, to study the structure of finite groups, it is enough to understand groups with no proper normal subgroup (that is, one that is neither the group itself nor the subgroup consisting only of the identity element): these are called simple groups.
