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The location of quasinormal subgroups in a group is not particularly well known. Maximal ones always have to be normal, but little has been proved about the minimal ones. In finite groups, the difficulties arise in the p-groups. Here we prove that, for every odd prime p, a quasinormal subgroup of order p2 in a finite p-group G contains a quasinormal subgroup of G of order p.
By a well-known result of Green (Proc R Soc A 237:574–581, 1956) and the formal definition of Ellis and Wiegold (Bull Austral Math Soc 60:191–196, 1999), there is an integer t, say corank(G), such that $${|\mathcal{M}(G)| = p^{\frac{1}{2}n(n-1)-t}}$$ . In Niroomand (J Algebra 322:4479–4482, 2009), the author showed for a non-abelian group G, corank(G) ≥ logp(|G|)−2 and classified the structure...
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