Groups in Which All Subgroups of Infinite Rank Are Subnormal

Abstract

Abstract. Let G be a locally soluble-by-finite group in which every nonsubnormal subgroup has finite rank. It is proved that either G has finite rank or G is soluble and locally nilpotent (and even a Baer group). On the other hand, a group G is constructed that has infinite rank and satisfies the given hypothesis, but does not have every subgroup subnormal.2000 Mathematics Subject Classification. 20E15, 20F19.

On groups whose subgroups of infinite rank are Sylow permutable

“…The structure of quasihamiltonian groups have been completely described by Iwasawa (see [14,Chapter 2]): they are exactly the locally nilpotent groups with modular subgroup lattice. Dixon and Karatas ( [11]), motivated by the papers [12,13], proved that if a (generalised) soluble group G of infinite special rank has all its subgroups of infinite special rank permutable, then G is quasihamiltonian.…”

“…The results of the present paper analyse the impact of the embedding of the subgroups of infinite section rank on the structure of a periodic group and can be viewed as belonging to a larger family of results stating that in a group of infinite special rank the behaviour of subgroups of finite special rank with respect to a given subgroup embedding property can be ignored (see [4][5][6][7][8][10][11][12][13]). …”

Groups of Infinite Rank With Normality Conditions on Subgroups With Small Normal closure

“…In particular, it follows that every proper subnormal subgroup of infinite rank of a SC ∞ -group is soluble (see [9,Theorem 2]). Theorem 3.2.…”

Groups with small deviation for non-subnormal subgroups

“…Since groups with subgroups of defect at most are known to be nilpotent [18], while groups with all subgroups subnormal need not be [6], it is to be expected that this further restriction is a strong one. In this regard, the main result of [2] might be compared with those of [10]. Our principal results here are concerned with groups G satisfying − (G) ≤ 1 that are either locally nilpotent or soluble.…”