Finite core-pp-groups

Abstract

Forna positive integer, a groupGis calledcore-nifH/H Ghas order at mostnfor every subgroupHofG(whereH Gis the normal core ofH, the largest normal subgroup ofGcontained inH). It is proved that a finite core-pp-groupGhas a normal abelian subgroup whose index inGis at mostp 2ifpâ� 2, which is the best possible bound, and at most 2 6ifp=2. © 1997 Academic Press

More on core-2 2-groups

“…Subgroups of Q are core-free, and if jQj > p, then Q has a 2-generated subgroup of order greater than p, a contradiction. Another equivalent characterisation of core-p-ness may be found in [3,Lemma 1.3]: The group G enjoys core-p-ness if and only if every nontrivial subgroup H of G has a maximal subgroup M with OEM; G Ĉ.H /.…”

“…A group G is called core-n if jH=H G j Ä n for every subgroup H of G; here, H G denotes the largest normal subgroup of G contained in H . In [4], it was shown that a locally finite core-n group has a normal abelian subgroup whose index in the group is bounded by a function of n. In [3] the same authors proved that, for odd primes p, a finite core-p p-group has an abelian normal subgroup of index p 2 , a best possible bound. The paper ( [3,Theorem 2]) also contains a proof of the existence of a normal abelian subgroup of index at most 2 6 in a finite core-2 2-group.…”

“…In [4], it was shown that a locally finite core-n group has a normal abelian subgroup whose index in the group is bounded by a function of n. In [3] the same authors proved that, for odd primes p, a finite core-p p-group has an abelian normal subgroup of index p 2 , a best possible bound. The paper ( [3,Theorem 2]) also contains a proof of the existence of a normal abelian subgroup of index at most 2 6 in a finite core-2 2-group. 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.…”

“…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

“…In [2], Lennox, Smith and Wiegold show that, for p = 2, a core-p p-group is nilpotent of class at most 3 and has an abelian normal subgroup of index at most p 5 . 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.…”