Using Lagrange’s Theorem and Cauchy’s Theorem one may show that a finite group is a -group if and only if is a power of .
A -subgroup (or -primary subgroup) of a group is a subgroup (http://planetmath.org/Subgroup) of such that is also a -group. A group that is a -subgroup for some prime is also called a primary subgroup.
It follows from Zorn’s Lemma that every group has a maximal -subgroup, for every prime . The maximal -subgroup need not be unique (though for abelian groups it is always unique, and is called the -primary component of the abelian group). A maximal -subgroup may, of course, be trivial. Non-trivial maximal -subgroups of finite groups are called Sylow -subgroups (http://planetmath.org/SylowPSubgroups).
|Date of creation||2013-03-22 14:53:08|
|Last modified on||2013-03-22 14:53:08|
|Last modified by||yark (2760)|