Groups with all Subgroups Subnormal or Nilpotent-by-Chernikov

Abstract

It was proved by Asar in [1] that a locally nilpotent group in which every proper subgroup is nilpotent-by-Chernikov is itself nilpotent-by-Chernikov.It is easy to see that the main problem here lies with establishing solubility, since a non-trivial soluble group G has a homomorphic image that is either cyclic or a Pru Èfer p-group, and if M / N / G; with M nilpotent and both G=N and N=M Chernikov, then M G is easily shown to be nilpotent, and of course G=M G is Chernikov.In view of earlier results from [7], Asar's result completes the proof that a locally graded group with all proper subgroups nilpotent-by-Chernikov is nilpotent-by-Chernikov; in particular, a locally graded group with all proper subgroups nilpotent is soluble-by-finite.Now, by a well-known result due to Mo Èhres [6], a group G with all subgroups subnormal is soluble, and in [9] and [10] the question was considered as to what might be said about a group G in which every subgroup is either subnormal or nilpotent.With regard to establishing solubility, the main two results in [10] were that a locally soluble-byfinite group with all subgroups subnormal or nilpotent is itself soluble (Theorem 1), and a locally graded group in which every subgroup is either nilpotent or subnormal of defect at most n is also soluble (Theorem 2).(Let us note here that there is an error, but one that is easily corrected, in the proof of Lemma 2:2 of [10], in that the appeal to Lemma 1:2 at the beginning of the second paragraph really requires an amended version of Lemma 1:2; where solubility replaces nilpotency throughout; the details are almost identical.A similar remark applies to the proof of the Proposition in [9], which invokes Lemma 3 of that paper, but here an alternative correction is to include in the statement of

Locations

• Rendiconti del Seminario Matematico della Università di Padova - View - PDF
• French digital mathematics library (Numdam) - View - PDF