nLab finite group

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 $C_G(F^*(G))=Z(F^*(G))$ , where $F^*(G)$ is the generalized Fitting subgroup of $G$ , defined below, $C_G(F^*(G))$ is the subgroup of $G$ consisting of all elements commuting with every element of $F^*(G)$ , and $Z(H)$ for any group $H$ is the center of $H$ , the subgroup of $H$ consisting of all elements commuting with every element of $H$ . Thus $G$ is somehow assembled from $F^*(G)$ , whose structure has some easy features, and $G/C_G(F^*(G))$ , which is isomorphic to a subgroup of the automorphism group of $F^*(G)$ and which has a quotient group isomorphic to $G/F^*(G)$ .

One definition of $F^*(G)$ is that it is the subgroup generated by all normal subgroups $N$ of $G$ possessing subgroups $N_1,N_2,\dots, N_r$ for some integer $r$ such that $N=N_1N_2\cdots N_r$ ; $x_i x_j=x_j x_i$ for all $x_i\in N_i$ , $x_j\in N_j$ , and distinct subscripts $i$ and $j$ ; and each $N_i$ either has prime power order or is a quasisimple group. Bender proved that $F^*(G)$ itself enjoys these properties.

Finally a group $H$ is called quasisimple if and only if $H=[H,H]$ and $H/Z(H)$ 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

classification of finite simple groups.

“Most finite groups are nilpotent”

The meaning of the title is this curious fact (based on empirical evidence, anyway): if we are counting isomorphism classes of finite groups up to a given order of a grouporder?, then most of them are $2$ -primary groups (and therefore nilpotent; see at 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”.)

Discussion can be found here and here.

Examples

For every natural number $n \in \mathbb{N}$ , the cyclic group

$\mathbb{Z}_n := \mathbb{Z}/n \mathbb{Z}$

is finite.
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.
finite subgroups of SO(3) and finite subgroups of SU(2) have an ADE classification
the general linear group over prime fields are finit, such as GL(2,3)
wallpaper group
see also at finite abelian group.

ADE classification and McKay correspondence

Dynkin diagram/ Dynkin quiver	dihedron, Platonic solid	finite subgroups of SO(3)	finite subgroups of SU(2)	simple Lie group
$A_{n \geq 1}$		cyclic group $\mathbb{Z}_{n+1}$	cyclic group $\mathbb{Z}_{n+1}$	special unitary group $SU(n+1)$
A1		cyclic group of order 2 $\mathbb{Z}_2$	cyclic group of order 2 $\mathbb{Z}_2$	SU(2)
A2		cyclic group of order 3 $\mathbb{Z}_3$	cyclic group of order 3 $\mathbb{Z}_3$	SU(3)
A3 = D3		cyclic group of order 4 $\mathbb{Z}_4$	cyclic group of order 4 $2 D_2 \simeq \mathbb{Z}_4$	SU(4) $\simeq$ Spin(6)
D4	dihedron on bigon	Klein four-group $D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$	quaternion group $2 D_4 \simeq$ Q8	SO(8), Spin(8)
D5	dihedron on triangle	dihedral group of order 6 $D_6$	binary dihedral group of order 12 $2 D_6$	SO(10), Spin(10)
D6	dihedron on square	dihedral group of order 8 $D_8$	binary dihedral group of order 16 $2 D_{8}$	SO(12), Spin(12)
$D_{n \geq 4}$	dihedron, hosohedron	dihedral group $D_{2(n-2)}$	binary dihedral group $2 D_{2(n-2)}$	special orthogonal group, spin group $SO(2n)$ , $Spin(2n)$
$E_6$	tetrahedron	tetrahedral group $T$	binary tetrahedral group $2T$	E6
$E_7$	cube, octahedron	octahedral group $O$	binary octahedral group $2O$	E7
$E_8$	dodecahedron, icosahedron	icosahedral group $I$	binary icosahedral group $2I$	E8

Related concepts

References

Early textbook accounts:

William Burnside, Theory of Groups of Finite Order, 1897 (pdf), reprinted by Cambridge University Press 2012 (doi:10.1017/CBO9781139237253)
G. A. Miller, H. F. Blichfeldt, Leonard E. Dickson, Theory and applications of finite groups, Dover, New York, 1916 (ark:/13960/t0ht2kb4x)

Discussion in homotopy type theory/univalent foundations of mathematics:

Marc Bezem, Ulrik Buchholtz, Pierre Cagne, Bjørn Ian Dundas, Daniel R. Grayson: Chapter 7 of: Symmetry (2021) $[$ pdf $]$

On the representation theory (linear representations) of finite groups over algebraic number fields:

Louis Solomon, The representation of finite groups in algebraic number fields, J. Math. Soc. Japan Volume 13, Number 2 (1961), 144-164 (euclid:euclid.jmsj/1261147137)

Discussion of group characters and group cohomology of finite groups:

Michael Atiyah, Characters and cohomology of finite groups, Publications Mathématiques de l’IHÉS, 9 (1961), p. 23-64 (numdam:PMIHES_1961__9__23_0)
Alejandro Adem, R. James Milgram, Cohomology of Finite Groups, Springer 2004
Narthana Epa, Platonic 2-groups, 2010 (pdf)