Classical groups Finite groups Group schemes Topological groups Lie groups Super-Lie groups Higher groups Cohomology and Extensions Related concepts A finite group is a group whose underlying set is finite. This is equivalently a group object in FinSet. Let be a finite group with order . (Cauchy) If a prime number divides , then equivalently has an element of order ; has a subgroup of order of a group . See at Cauchy's theorem for more. See at Feit-Thompson theorem. The structure of finite groups is a very hard problem; the classification of finite simple groups alone is one of the largest theorems ever proved (certainly if measured by number of journal pages needed for a complete proof). All finite groups are built out of simple groups, but the ways to do this have not (yet?) been fully classified. A point of view that can be useful in particular cases – more useful than the Jordan-Hölder theorem – is provided by the F-theorem?, due to Hans Fitting in the solvable case and Helmut Bender in the general case. It states that , where is the generalized Fitting subgroup of , defined below, is the subgroup of consisting of all elements commuting with every element of , and for any group is the center of , the subgroup of consisting of all elements commuting with every element of . Thus is somehow assembled from , whose structure has some easy features, and , which is isomorphic to a subgroup of the automorphism group of and which has a quotient group isomorphic to . One definition of is that it is the subgroup generated by all normal subgroups of possessing subgroups for some integer such that ; for all , , and distinct subscripts and ; and each either has prime power order or is a quasisimple group. Bender proved that itself enjoys these properties. Finally a group is called quasisimple if and only if and is simple. The finite quasisimple groups have been classified, as a consequence of the classification of finite simple groups and the calculation of the Schur...