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Let ℨ be a complete set of Sylow subgroups of a group G. A subgroup H of G is called ℨ-permutably embedded in G if every Sylow subgroup of H is also a Sylow subgroup of some ℨ-permutable subgroup of G. By using this concept, we obtain some new criteria of p-supersolubility and p-nilpotency of a finite group.

Conjugacy classes in a p-group of maximal class of order p^8

Antonio Vera López, Gustavo A. Fernández Alcober (1989)

Extracta Mathematicae

Construction of finite $p$ -groups with prescribed group of noncentral automorphisms

Panagiotis C. Soules (1986)

Rendiconti del Seminario Matematico della Università di Padova

Converse of a Theorem of Mal'cev on Nilpotent Groups.

James Wiegold, John C. Lennox (1974)

Mathematische Zeitschrift

Counting $p$ -groups and nilpotent groups

Marcus Du Sautoy (2000)

Publications Mathématiques de l'IHÉS

Cyclic and dihedral constructions of even order

Aleš Drápal (2003)

Commentationes Mathematicae Universitatis Carolinae

Let $G (\circ)$ and $G (*)$ be two groups of finite order $n$ , and suppose that they share a normal subgroup $S$ such that $u \circ v = u * v$ if $u \in S$ or $v \in S$ . Cases when $G / S$ is cyclic or dihedral and when $u \circ v \neq u * v$ for exactly $n^{2} / 4$ pairs $(u, v) \in G \times G$ have been shown to be of crucial importance when studying pairs of 2-groups with the latter property. In such cases one can describe two general constructions how to get all possible $G (*)$ from a given $G = G (\circ)$ . The constructions, denoted by $G [α, h]$ and $G [β, γ, h]$ , respectively, depend on a coset $α$ (or two cosets $β$ and $γ$ ) modulo $S$ , and on an...

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