non-isomorphic groups of given order
Theorem. For every positive integer , there exists only a finite amount of non-isomorphic groups of order .
This assertion follows from Cayley’s theorem, according to which any group of order is isomorphic with a subgroup of the symmetric group . The number of non-isomorphic subgroups of cannot be greater than
The above theorem may be used in proving the following Landau’s theorem:
Theorem (Landau). For every positive integer , there exists only a finite amount of finite non-isomorphic groups which contain exactly conjugacy classes of elements.
One needs also the
Lemma. If and , then there is at most a finite amount of the vectors consisting of positive integers such that
The lemma is easily proved by induction on .
|Title||non-isomorphic groups of given order|
|Date of creation||2013-03-22 18:56:38|
|Last modified on||2013-03-22 18:56:38|
|Last modified by||pahio (2872)|