non-isomorphic groups of given order


Theorem.  For every positive integer n, there exists only a finite amount of non-isomorphic groups of order n.

This assertion follows from Cayley’s theorem, according to which any group of order n is isomorphicPlanetmathPlanetmathPlanetmathPlanetmath with a subgroupMathworldPlanetmathPlanetmath of the symmetric groupMathworldPlanetmathPlanetmath 𝔖n.  The number of non-isomorphic subgroups of 𝔖n cannot be greater than

(n!-1n-1).

The above theorem may be used in proving the following Landau’s theorem:

Theorem (Landau).  For every positive integer n, there exists only a finite amount of finite non-isomorphic groups which contain exactly n conjugacy classesMathworldPlanetmathPlanetmath of elements.

One needs also the

Lemma.  If  n+  and  0<r,  then there is at most a finite amount of the vectors  (m1,m2,,mn)  consisting of positive integers such that

j=1n1mj=r.

The lemma is easily proved by induction on n.

Title non-isomorphic groups of given order
Canonical name NonisomorphicGroupsOfGivenOrder
Date of creation 2013-03-22 18:56:38
Last modified on 2013-03-22 18:56:38
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Theorem
Classification msc 20A05
Related topic BinomialCoefficient
Related topic PropertiesOfConjugacy
Defines Landau’s theorem