Omega subgroups not are variety-containing

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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)
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Statement

We can have a group of prime power order (i.e., a finite p-group) P such that the first omega subgroup \Omega_1(P), defined as:

\Omega_1(P) := \langle x \mid x^p = e \rangle,

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 \Omega_1(P) but that is not contained in \Omega_1(P).

In particular, because of the equivalence of definitions of variety-containing subgroup of finite group, \Omega_1(P) need not be a Subisomorph-containing subgroup (?) and it need not be a Variety-containing subgroup (?).

Related facts

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, \Omega_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 \Omega_1(P), but H itself is not contained in \Omega_1(P).