Omega subgroups are variety-containing in regular p-group

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Definition

Suppose P is a Regular p-group (?). Then, for any natural number k, the subgroup:

\Omega_k(P) := \langle x \mid x^{p^k} = e \rangle

is a Variety-containing subgroup (?) of P (i.e., a Variety-containing subgroup of group of prime power order (?)): it contains any subgroup of P that is in the subvariety of the variety of groups generated by \Omega_k(P).

In particular, \Omega_k(P) is a Subhomomorph-containing subgroup (?) and a Subisomorph-containing subgroup (?) of P.

Related facts

Facts used

  1. Omega subgroup equals set of elements of the exponent dividing the prime power in regular p-group

Proof

By fact (1), \Omega_k(P) is precisely the subset of P comprising the elements of P whose order divides p^k.

Let \mathcal{V} be the variety of all groups in which the order of every element divides p^k. Note that this is a variety and it contains \Omega_k(P). Thus, the subvariety generated by \Omega_k(P) is contained in \mathcal{V}. Further, any element of \mathcal{V} that is a subgroup of P is contained in \Omega_k(P). Thus, \Omega_k(P) contains all subgroups of P in the subvariety it generates.