Let be a finite group with order .
If a prime number divides , then equivalently
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 multiplier of each finite simple group.
For more on this see
The meaning of the title is this curious fact (based on empirical evidence, anyway): if we are counting isomorphism classes of groups up to a given order, then most of them are -primary groups (and therefore nilpotent; see class equation). For example, it is reported that “out of the 49,910,529,484 groups of order at most 2000, a staggering 49,487,365,422 of them have order 1024”.
(It has also been suggested that a more meaningful weighting would divide each isomorphism class representative by the order of its automorphism group; as of this writing the nLab authors don’t know how much this would affect the strength of the assertion “most finite groups are nilpotent”.)
The largest finite group that is also a sporadic simple group, i.e., does not belong(up to isomorphism) to the infinite family of the alternating groups or to the infinite family of finite groups of Lie type, is the Monster group.
For more see also at finite abelian group.
|Dynkin diagram||Platonic solid||finite subgroup of||finite subgroup of||simple Lie group|
|cyclic group||cyclic group||special unitary group|
|dihedron/hosohedron||dihedral group||binary dihedral group||special orthogonal group|
|tetrahedron||tetrahedral group||binary tetrahedral group||E6|
|cube/octahedron||octahedral group||binary octahedral group||E7|
|dodecahedron/icosahedron||icosahedral group||binary icosahedral group||E8|
Alejandro Adem, R.James Milgram, Cohomology of Finite Groups, Springer 2004
Adam Keenan, Which finite groups act freely on spheres?, 2003 (pdf)