On Core-2 Groups

More on core-2 2-groups

“…The authors mention that no examples of finite core-2 2-groups were known that did not possess an abelian subgroup of index at most 4. Later ( [5]) it was shown that finite core-2 groups of class 2 indeed must have an abelian subgroup of index at most 4. The present paper is devoted to showing that every finite core-2 2-group has an abelian subgroup of index at most 4.…”

“…The assertions of the next lemma, describing finite core-p p-groups G witĥ .G/ Â Z.G/, are partly contained in [3,Theorem 1] and [5,Theorem]. We shall, however, require more detail than is provided in the referenced sources.…”

On Finite Quasi-Core-p p-Groups

“…Take g ∈ G with g 4 / ∈ G . Then [g 2 , h] ∈ Ω 1 ( g ) for any h ∈ G, which implies [g 4 , h] = 1 and therefore g 4 ∈ Z(G). So 2 (G) ≤ G Z(G).…”

“…Furthermore, Cutolo, Khukhro, Lennox, Wiegold, Rinauro and Smith [3] prove that a core-p p-group G has a normal abelian subgroup whose index in G is at most p 2 if p = 2. Furthermore, if p = 2, Cutolo, Smith and Wiegold [4] prove that every core-2 2-group has an abelian subgroup of index at most 16. As a deepening of research in this area, it is interesting to study the following question.…”

Finite -Groups All of Whose Nonnormal Subgroups Have Bounded Normal Cores

“…In [8] and [10], Lv et al studied finite p-groups G with |H G : H| ≤ p 2 for every nonnormal subgroup H of G and showed that the best upper bound for the nilpotent class of such groups is This work was supported by the National Natural Science Foundation of China (11671324, 11471266). c 2019 Australian Mathematical Publishing Association Inc. 4 for p ≥ 3 and 5 for p = 2. Suppose now that H/H G has bounded order for each nonnormal subgroup H of a group G. In [2], it is proved that such a locally finite group is abelian-by-finite.…”