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
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View subgroup property dissatisfactions for subgroup-defining functions

Statement

We can have a group of prime power order (i.e., a finite $p$ -group) $P$ such that the first omega subgroup $Ω_{1} (P)$ , defined as:

$Ω_{1} (P) : = ⟨ x ∣ x^{p} = e ⟩$ ,

is not a variety-containing subgroup (i.e., Variety-containing subgroup of group of prime power order (?)) of $P$ : there exists a subgroup $H$ of $P$ isomorphic to a subgroup of $Ω_{1} (P)$ but that is not contained in $Ω_{1} (P)$ .

In particular, because of the equivalence of definitions of variety-containing subgroup of finite group, $Ω_{1} (P)$ need not be a Subisomorph-containing subgroup (?) and it need not be a Variety-containing subgroup (?).

Related facts

Omega subgroups are homomorph-containing

Proof

Further information: wreath product of groups of order p

Suppose $A$ is a wreath product of groups of order p, i.e., $A$ is a group of order $p^{p + 1}$ obtained as the semidirect product of an elementary abelian group of order $p^{p}$ by a cyclic group of order $p$ acting as automorphisms. $A$ is isomorphic to the $p$ -Sylow subgroup of the symmetric group of degree $p^{2}$ . In particular, $A$ has a cyclic subgroup of order $p^{2}$ .

Suppose $B$ is a cyclic group of order $p^{2}$ .

Define:

$P : = A \times B$ .

Then, $Ω_{1} (P) = A \times C$ where $C$ is the subgroup of order $p$ in $B$ . Consider the subgroup $H = {e} \times B$ . $H$ is isomorphic to a subgroup of order $p^{2}$ in $A \times {e}$ , which in turn is contained in $Ω_{1} (P)$ , but $H$ itself is not contained in $Ω_{1} (P)$ .