On automorphism groups of some finite groups

We show that if n > 1 is odd and has no divisor p4 for any prime p, then there is no finite group G such that│Aut(G)│ = n.     

Endomorphisms of automorphism groups of free groups

It is proved that any non-trivial endomorphism of an automorphism group AutF_n of a free group F_n, for n 3, either is an automorphism or factorization over a proper automorphism subgroup. An endomorphism of AutF₂ is an automorphism, or else a homomorphism onto one of the groups S₃, D₈, Z₂ × Z₂, Z₂, or (Z₂ × Z₂). A non-trivial homomorphism of AutF_n into AutF_m, for n 3, m 2, and n > m, is a homomorphism onto Z₂ with kernel SAutF_n. As a consequence, we obtain that AutF_n is co-Hopfian.Supported by RFBR grant No. 02-01-00293 and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 211–237, March–April, 2005.     

On mixed Abelian groups with periodic groups of automorphisms

In the paper, sufficient conditions for the splittability of mixed Abelian groups with periodic automorphism groups are established. Classes of mixed splittable Abelian groups with perfect holomorphs are distinguished. Translated fromMaternaticheskie Zametki, Vol. 61, No. 4, pp. 483–493, April, 1997. Translated by A. I. Shtern     

Characterization of automorphism groups of sporadic simple groups

It is proved that all automorphism groups of the sporadic simple groups are characterized by their element orders and the group orders.     

Characterization of generalized Chernikov groups among groups with involutions

The class of generalized Chernikov groups is characterized, i.e., the class of periodic locally solvable groups with the primary ascending chain condition. The name of the class is related to the fact that the structure of such groups is close to that of Chernikov groups. Namely, a Chernikov group is defined as a finite extension of a direct product of finitely many quasi-cyclic groups, and a generalized Chernikov group is a layer-finite extension of a direct productA of quasi-cyclicp-groups with finitely many factors for each primep such that each of its elements does not commute elementwise with only finitely many Sylow subgroups ofA. A theorem that characterizes the generalized Chernikov groups in the class of groups with involution is proved. Translated fromMatematicheskie Zametki, Vol. 62, No 4, pp. 577–587, October, 1997. Translated by A. I. Shtern     

On outer automorphism groups of coxeter groups

Summary It is shown that the outer automorphism group of a Coxeter groupW of finite rank is finite if the Coxeter graph contains no infinite bonds. A key step in the proof is to show that if the group is irreducible andΠ ₁ andΠ ₂ any two bases of the root system ofW, thenΠ ₂ = ±ωΠ ₁ for some ω εW. The proof of this latter fact employs some properties of the dominance order on the root system introduced by Brink and Howlett. This article was processed by the author using the Springer-Verlag TEX PJour1g macro package 1991.     

On representations of affine Kac-Moody groups and related loop groups

We demonstrate a one to one correspondence between the irreducible projective representations of an affine Kac-Moody group and those of the related loop group, which leads to the results that every non-trivial representation of an affine Kac-Moody group must have its degree greater than or equal to the rank of the group and that the equivalence appears if and only if the group is of type for some . Moreover the characteristics of the base fields for the non-trivial representations are found being always zero.

     

On hypercentral groups

Let G be a hypercentral group. Our main result here is that if G/G’ is divisible by finite then G itself is divisible by finite. This extends a recent result of Heng, Duan and Chen [2], who prove in a slightly weaker form the special case where G is also a p-group. If G is torsion-free, then G is actually divisible.      

Divisible rigid groups

A soluble group G is rigid if it contains a normal series of the form G = G₁ > G₂ > … > G_p > G_p+1 = 1, whose quotients G_i/G_i+1 are Abelian and are torsion-free as right ℤ[G/G_i]-modules. The concept of a rigid group appeared in studying algebraic geometry over groups that are close to free soluble. In the class of all rigid groups, we distinguish divisible groups the elements of whose quotients G_i/G_i+1 are divisible by any elements of respective groups rings Z[G/G_i]. It is reasonable to suppose that algebraic geometry over divisible rigid groups is rather well structured. Abstract properties of such groups are investigated. It is proved that in every divisible rigid group H that contains G as a subgroup, there is a minimal divisible subgroup including G, which we call a divisible closure of G in H. Among divisible closures of G are divisible completions of G that are distinguished by some natural condition. It is shown that a divisible completion is defined uniquely up to G-isomorphism. Supported by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-344.2008.1). Translated from Algebra i Logika, Vol. 47, No. 6, pp. 762–776, November–December, 2008.     

The abelianization of hypercyclic groups

Let G be a hypercyclic group. The most substantial results of this paper are the following. a) If G/G′ is 2-divisible, then G is 2-divisible. b) If G/G′ is a 2′-group, then G is a 2′-group. c) If G/G′ is divisible by finite-of-odd-order, then G/V is divisible by finite-of-odd-order, where V is the intersection of the lower central series (continued transfinitely) of O _2′ (G).      

Mordell-Weil groups and Selmer groups of twin- prime elliptic curves

Let E = Eσ : y2 = x(x + σp)(x + σq) be elliptic curves, where σ = ±1, p and q are primenumbers with p+2 = q. (i) Selmer groups S(2)(E/Q), S(φ)(E/Q), and S(φ)(E/Q) are explicitly determined,e.g. S(2)(E+1/Q)= (Z/2Z)2, (Z/2Z)3, and (Z/2Z)4 when p ≡ 5, 1 (or 3), and 7(mod 8), respectively. (ii)When p ≡ 5 (3, 5 for σ = -1) (mod 8), it is proved that the Mordell-Weil group E(Q) ≌ Z/2Z Z/2Z,symbol, the torsion subgroup E(K)tors for any number field K, etc. are also obtained.     

Tychonoff property for linear groups

A criterion for a wide class of topological groups which includes linear discrete groups and Lie groups to be Tychonoff groups is established. The main result provides a criterion for an almost polycyclic group to have the Tychonoff property. By the well-known Tits alternative, this yields the required criterion for linear discrete groups. In conclusion it is pointed out that a particular case of the presented proof yields a Tychonoff property criterion for Lie groups. In addition, an example of a polycyclic group without Tychonoff subgroups of finite index is constructed. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 269–279, February, 1998. The author wishes to express his gratitude to R. I. Grigorchuk for setting the problem and his interest in the work. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00182 and by the American Mathematical Society Fund.     

Hyperbolic unit groups

In this paper we study the groups whose integral group rings have hyperbolic unit groups . We classify completely the torsion subgroups of and the polycyclic-by-finite subgroups of the group . Finally, we classify the groups for which the boundary of has dimension zero.

     

On amenability of two-generator groups

A discrete group G is amenable if there exists a finitely additive probability measure on G which is invariant under left translations and is defined on all subsets of G. It is proved that if the group is generated by two elements and is amenable then there are words being relators whose most of the consecutive pairs of the letters belong to a certain four-element set of pairs. This fact is applied to reproving non-amenability of a braid group. The same group provides an example showing that such type of condition is not su?cient for amenabilty.     

Picard groups of multiplicative invariants

Let S = kA denote the group algebra of a finitely generated free abelian group A over the field k and let G be a finite subgroup of GL(A). Then G acts on S by means of the unique extension of the natural GL(A)-action on A. We determine the Picard group Pic R of the algebra of invariants R = S ^G . As an application, we produce new polycyclic group algebras with nontrivial torsion in K ₀ . Received: April 25, 1996     

A characterization of alternating groups

It is proved that a group G generated by a conjugacy class X of elements of order 3, so that every two non-commuting elements of X generate a subgroup isomorphic to an alternating group of degree 4 or 5, is locally finite. More precisely, either G contains a normal elementary 2-subgroup of index 3, or G is isomorphic to an alternating group of permutations on some (possibly infinite) set.Supported by RFBR grant Nos. 02-01-00495 and 02-01-39005, by FP Universities of Russia grant No. UR.04.01.0202, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1.Translated from Algebra i Logika, Vol. 44, No. 1, pp. 54–69, January–February, 2005.     

Branch groups

The class of branch groups is defined (both in the abstract and in the profinite category). The relationship of this class with the class of extremal groups is established. Properties of the branch groups are investigated. Applications of the congruence property to the theory of profinite branch groups are indicated. The weak maximality of parabolic subgroups in branch groups is proved. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 852–858, June, 2000.     

Spectra of finite linear and unitary groups

The spectrum of a finite group is the set of its element orders. An arithmetic criterion determining whether a given natural number belongs to a spectrum of a given group is furnished for all finite special, projective general, and projective special linear and unitary groups. Supported by RFBR (grant Nos. 08-01-00322 and 06-01-39001) and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (project NSh-344.2008.1). __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 157–173, March–April, 2008.     

Embedding right-angled Artin groups into graph braid groups

We construct an embedding of any right-angled Artin group G(Δ) defined by a graph Δ into a graph braid group. The number of strands required for the braid group is equal to the chromatic number of Δ. This construction yields an example of a hyperbolic surface subgroup embedded in a two strand planar graph braid group.