Omega subgroups not are variety-containing
This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., omega subgroups of group of prime power order) does not always satisfy a particular subgroup property (i.e., variety-containing subgroup)
View subgroup property satisfactions for subgroup-defining functions View subgroup property dissatisfactions for subgroup-defining functions
is not a variety-containing subgroup (i.e., Variety-containing subgroup of group of prime power order (?)) of : there exists a subgroup of isomorphic to a subgroup of but that is not contained in .
In particular, because of the equivalence of definitions of variety-containing subgroup of finite group, need not be a Subisomorph-containing subgroup (?) and it need not be a Variety-containing subgroup (?).
Further information: wreath product of groups of order p
Suppose is a wreath product of groups of order p, i.e., is a group of order obtained as the semidirect product of an elementary abelian group of order by a cyclic group of order acting as automorphisms. is isomorphic to the -Sylow subgroup of the symmetric group of degree . In particular, has a cyclic subgroup of order .
Suppose is a cyclic group of order .
Then, where is the subgroup of order in . Consider the subgroup . is isomorphic to a subgroup of order in , which in turn is contained in , but itself is not contained in .