nLab
finite group

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 GG be a finite group with order |G|{\vert G\vert} \in \mathbb{N}.

Theorem

(Cauchy)

If a prime number pp divides |G|{\vert G\vert}, then equivalently

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

One definition of F *(G)F^*(G) is that it is the subgroup generated by all normal subgroups NN of GG possessing subgroups N 1,N 2,,N rN_1,N_2,\dots, N_r for some integer rr such that N=N 1N 2N rN=N_1N_2\cdots N_r; x ix j=x jx ix_i x_j=x_j x_i for all x iN ix_i\in N_i, x jN jx_j\in N_j, and distinct subscripts ii and jj; and each N iN_i either has prime power order or is a quasisimple group. Bender proved that F *(G)F^*(G) itself enjoys these properties.

Finally a group HH is called quasisimple if and only if H=[H,H]H=[H,H] and H/Z(H)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 22-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

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

    n:=/n \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.

ADE classification

Dynkin diagramPlatonic solidfinite subgroup of SO(3)SO(3)simple Lie group
A lA_lcyclic groupspecial unitary group
D lD_ldihedron/hosohedrondihedral groupspecial orthogonal group
E 6E_6tetrahedrontetrahedral groupE6
E 7E_7cube/octahedronoctahedral groupE7
E 8E_8dodecahedron/icosahedronicosahedral groupE8

References

Discussion of group characters and group cohomology of finite groups includes

  • 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)

Discussion of free actions of finite groups on n-spheres (see also at ADE classification) includes

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

  • Adam Keenan, Which finite groups act freely on spheres?, 2003 (pdf)

Revised on August 19, 2015 09:35:02 by Urs Schreiber (131.227.23.37)