Contents

group theory

# Contents

## Definition

A finite group is a group whose underlying set is finite.

This is equivalently a group object in FinSet.

## Properties

### Cauchy’s theorem

Let $G$ be a finite group with order ${\vert G\vert} \in \mathbb{N}$.

###### Theorem

(Cauchy)

If a prime number $p$ divides ${\vert G\vert}$, then equivalently

• $G$ has an element of order $p$;

• $G$ has a subgroup of order of a group $p$.

See at Cauchy's theorem for more.

### Feit-Thompson theorem

###### Theorem

Every finite group of odd order is a solvable group.

See at Feit-Thompson theorem.

### Classification

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

#### “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 groups up to a given order, then most of them are $2$-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”.)

Discussion can be found here and here.

## Examples

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
$SU(n+1)$
A1cyclic group of order 2
$\mathbb{Z}_2$
cyclic group of order 2
$\mathbb{Z}_2$
SU(2)
A2cyclic 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)
D4dihedron 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)
D5dihedron on
triangle
dihedral group of order 6
$D_6$
binary dihedral group of order 12
$2 D_6$
SO(10), Spin(10)
D6dihedron 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
$SO(2n)$, $Spin(2n)$
$E_6$tetrahedrontetrahedral 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

An early textbook on finite groups:

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

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)

• Alejandro Adem, R.James Milgram, Cohomology of Finite Groups, Springer 2004

• Narthana Epa, Platonic 2-groups, 2010 (pdf)

• John Milnor, Groups which act on $S^n$ without fixed points, American Journal of Mathematics Vol. 79, No. 3 (Jul., 1957), pp. 623-630 (JSTOR)