Groups with finitely many derived subgroups of non-normal subgroups

A group is called metahamiltonian if all its non-abelian subgroups are normal; it is known that locally soluble metahamiltonian groups have finite derived subgroup. This result is generalized here, by proving that every locally graded group with finitely many derived subgroups of non-normal subgroups has finite derived subgroup. Moreover, locally graded groups having only finitely many derived subgroups of infinite non-normal subgroups are completely described. Received: 25 April 2005     

Groups Covered by an Infinite Number of Abelian Subgroups

If a group G is covered by Abelian subgroups ( an infinite cardinal) then and this estimate is sharp. This answers a question of Faber, Laver and McKenzie. Received February 1, 2000     

Nilpotent Subgroups of Groups with All Subgroups Subnormal

It is proved that in groups with all subgroups subnormal everynilpotent subgroup is contained in a normal nilpotent subgroup.     

Groups with many periodic factor-groups

Groups, all proper factor-groups of which are hyperfmite, are studied in this article.     

Groups with a positive law on commutators

Let p be a prime number and let G be a finitely generated group that is residually a finite p-group. We prove that if G satisfies a positive law on all elements of the form [a,b][c,d]ⁱ, a,b,c,d∈G and i?0, then the entire derived subgroup G^′ satisfies a positive law. In fact, G^′ is an extension of a nilpotent group by a locally finite group of finite exponent.     

Groups with a maximality condition for some non-normal subgroups

We describe (generalized) soluble-by-finite groups in which the set of non-normal subgroups which are not finitely generated satisfies the maximal condition.     

Locally Finite Groups with All Subgroups Normal-by-(Finite Rank)

A group is said to have finite (special) rank ≤ sif all of its finitely generated subgroups can be generated byselements. LetGbe a locally finite group and suppose thatH/H_Ghas finite rank for all subgroupsHofG, whereH_Gdenotes the normal core ofHinG. We prove that thenGhas an abelian normal subgroup whose quotient is of finite rank (Theorem 5). If, in addition, there is a finite numberrbounding all of the ranks ofH/H_G, thenGhas an abelian subgroup whose quotient is of finite rank bounded in terms ofronly (Theorem 4). These results are based on analogous theorems on locally finitep-groups, in which case the groupGis also abelian-by-finite (Theorems 2 and 3).     

Commutators in Residually Finite Groups

 The following result is proved. Let G be a residually finite group satisfying the identity ([x ₁, x ₂][x ₃, x ₄]) ⁿ  ≡ 1 for a positive integer n that is not divisible by p ² q ² for any distinct primes p and q. Then G′ is locally finite. Received 7 May 2001; in revised form 3 December 2001     

On Abelian Groups Having All Proper Fully Invariant Subgroups Isomorphic

We introduce two classes of abelian groups which have either only trivial fully invariant subgroups or all their nontrivial (respectively nonzero) fully invariant subgroups are isomorphic, called IFI-groups and strongly IFI-groups, such that every strongly IFI-group is an IFI-group, respectively. Moreover, these classes coincide when the groups are torsion-free, but are different when the groups are torsion as well as, surprisingly, mixed groups cannot be IFI-groups. We also study their important properties as our results somewhat contrast with those from [13 Grinshpon , S. Ya. , Nikolskaya (Savinkova) , M. M. ( 2011 ). Fully invariant subgroups of abelian p-groups with finite Ulm-Kaplansky invariants . Commun. Algebra 39 : 4273 – 4282 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] and [14 Grinshpon , S. Ya. , Nikolskaya (Savinkova) , M. M. ( 2011-2012/2014 ). Torsion IF-groups . Fundam. Prikl. Mat. 17 : 47 – 58 ; translated in J. Math. Sci. 197:614–622 . [Google Scholar]].     

The Hopf Property for Subgroups of Hyperbolic Groups

A group is said to be Hopfian if every surjective endomorphism of the group is injective. We show that finitely generated subgroups of torsion-free hyperbolic groups are Hopfian. Our proof generalizes a theorem of Sela (Topology 35 (2) 1999, 301–321).     

Engel Values in Residually Finite Groups

The following theorem is proved. Let n be a positive integer and q a power of a prime p. There exists a number m = m(n, q) depending only on n and q such that if G is any residually finite group satisfying the identity ([x _1,n y ₁] ⋯ [x _m,n y _m ])^q ≡ 1, then the verbal subgroup of G corresponding to the nth Engel word is locally finite.     

Characters of Quantum Groups of Type A n

A presentation for an arbitrary group extension is well known. A generalization of the work by Conway et al. (Group Tensor1972, 25, 405–418) on central extensions has been given by Baik et al. (J. Group Theor.). As an application of this we discuss necessary and sufficient conditions for the presentation of the central extension to be p-Cockcroft, where p is a prime or 0. Finally, we present some examples of this result.     

Groups with Few Nonmodular Subgroups

Let G be a Tarski-free group such that the join of all nonmodular subgroups of G is a proper subgroup in G. It is proved that G contains a finite normal subgroup N such that the quotient group G/N has a modular subgroup lattice.__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 10, pp. 1419 – 1423, October, 2004.     

Groups with Polycyclic Non-Normal Subgroups

The structure of groups in which many subgroups have a certain property X has been investigated for several choices of the property X. Groups whose non-normal subgroups satisfy certain finite rank conditions are studied in this article. In particular, a classification of groups in which every subgroup is either normal or polycyclic is given.(Dedicated to Mario Curzio on the occasion of his 70th birthday)1991 Mathematics Subject Classification: 20F16     

二次极大子群皆是$PSC^{*}$-群的有限群

This paper discusses the influence of minimal subgroups on the structure of finite groups and gives the structures of finite groups all of whose second maximal subgroups are PSC^＊-groups.     

Soluble groups with their centralizer factor groups of bounded rank

For a group class X, a group G is said to be a CX-group if the factor group G/C_G(g^G)∈X for all g∈G, where C_G(g^G) is the centralizer in G of the normal closure of g in G. For the class F_f of groups of finite order less than or equal to f, a classical result of B.H. Neumann [Groups with finite classes of conjugate elements, Proc. London Math. Soc. 1 (1951) 178-187] states that if G∈CF_f, the commutator group G^′ belongs to F_f^′ for some f^′ depending only on f. We prove that a similar result holds for the class , the class of soluble groups of derived length at most d which have Prüfer rank at most r. Namely, if , then for some r^′ depending only on r. Moreover, if , then for some r^′ and f^′ depending only on r,d and f.     

Products of Subgroups Which Are Subgroups

We prove conditions for a product of distinct subgroups of an arbitrary group G to be a subgroup of G. In particular, the normal closure of any A ≤ G is equal to the product of some distinct conjugates of A. As an application of the later result we derive constraints on the size of a nontrivial conjugacy class of a finite non-Abelian simple group.     

On the Non-existence of Semiregular Relative Difference Sets in Groups with All Sylow Subgroups Cyclic

We show that a group with all Sylow subgroups cyclic (other than ) cannot contain a normal semiregular relative difference set (RDSs). We also give a new proof that dihedral groups cannot contain (normal) semiregular RDSs either.