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Groupprops β

Omega subgroups are homomorph-containing

This article gives the statement, and possibly proof, of the fact that for any group, the subgroup obtained by applying a given subgroup-defining function (i.e., omega subgroups of group of prime power order) always satisfies a particular subgroup property (i.e., homomorph-containing subgroup)}
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Statement

Suppose G is a group of prime power order (i.e., a finite p-group for some prime number p). Then, the omega subgroups of G, defined as:

\Omega_j(G) := \langle x \mid x^{p^j} = e \rangle

are homomorph-containing subgroups of G.

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