nLab
finite group

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 $\mid G\mid \in \mathbb{N}$.

Theorem

(Cauchy)

If a prime number $p$ divides $\mid G\mid$, 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 $N=N_1{N}_2\cdots N_r$; $x_i{x}_j=x_j{x}_i$ 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.

Examples

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

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

• For more see also at finite abelian group.

Related concepts

• finite abelian group

References

• Atlas of finite group representations