In this paper we prove that if V is a vector space over a field of positive characteristric p ≠ 5 then any regular subgroup A of exponent 5 of GL(V) is cyclic. As a consequence a conjecture of Gupta and Mazurov is proved to be true.

In this paper we consider groups in which every subgroup has finite index in the nth term of its normal closure series, for a fixed integer n. We prove that such a group is the extension of a finite normal subgroup by a nilpotent group, whose class is bounded in terms of n only, provided it is either periodic or torsion-free.

In the present paper we consider Fitting classes of finite soluble groups which locally satisfy additional conditions related to the behaviour of their injectors. More precisely, we study Fitting classes 1 ≠⊆such that an-injector of G is, respectively, a normal, (sub)modular, normally embedded, system permutable subgroup of G for all G ∈.

Locally normal Fitting classes were studied before by various authors. Here we prove that some important results—already known for normality—are valid for all of the above mentioned embedding properties. For instance, all these embedding properties behave nicely with respect to the Lockett section. Further, for all of these properties the class of all finite soluble groups G such that an x-injector of G has the corresponding embedding property is not closed under forming normal products, and thus can fail to be a Fitting class.

Generalizing and strengthening some well-known results of Higman, B. Neumann, Hanna Neumann and Dark on embeddings into two-generator groups, we introduce a construction of subnormal verbal embedding of an arbitrary (soluble, fully ordered or torsion free) ordered countable group into a twogenerator ordered group with these properties. Further, we establish subnormal verbal embedding of defect two of an arbitrary (soluble, fully ordered or torsion free) ordered group G into a group with these properties and of the same cardinality as G, and show in connection with a problem of Heineken that the defect of such an embedding cannot be made smaller, that is, such verbal embeddings of ordered groups cannot in general be normal.

If ω ≡ 1 is a group law implying virtual nilpotence in every finitely generated metabelian group satisfying it, then it implies virtual nilpotence for the finitely generated groups of a large class of groups including all residually or locally soluble-or-finite groups. In fact the groups of satisfying such a law are all nilpotent-by-finite exponent where the nilpotency class and exponent in question are both bounded above in terms of the length of ω alone. This yields a dichotomy for words. Finally, if the law ω ≡ 1 satisfies a certain additional condition—obtaining in particular for any monoidal or Engel law—then the conclusion extends to the much larger class consisting of all ‘locally graded’ groups.

Let and denote respectively the variety of groups of exponent dividing e, the variety of nilpotent groups of class at most c, the class of nilpotent groups and the class of finite groups. It follows from a result due to Kargapolov and Čurkin and independently to Groves that in a variety not containing all metabelian groups, each polycyclic group G belongs to . We show that G is in fact in , where c is an integer depending only on the variety. On the other hand, it is not always possible to find an integer e (depending only on the variety) such that G belongs to but we characterize the varieties in which that is possible. In this case, there exists a function f such that, if G is d-generated, then G ∈ So, when e = 1, we obtain an extension of Zel'manov's result about the restricted Burnside problem (as one might expect, this result is used in our proof). Finally, we show that the class of locally nilpotent groups of a variety forms a variety if and only if for some integers c′, e′.

The graph product of a family of groups lies somewhere between their direct and free products, with the graph determining which pairs of groups commute. We show that the graph product of quasi-lattice ordered groups is quasi-lattice ordered, and, when the underlying groups are amenable, that it satisfies Nica's amenability condition for quasi-lattice orders. The associated Toeplitz algebras have a universal property, and their representations are faithful if the generating isometries satisfy a joint properness condition. When applied to right-angled Artin groups this yields a uniqueness theorem for the C*-algebra generated by a collection of isometries such that any two of them either *-commute or else have orthogonal ranges. The analogous result fails to hold for the nonabelian Artin groups of finite type considered by Brieskorn and Saito, and Deligne.

We provide an upper bound for the order of a nilpotent injector of a finite solvable group with Fitting subgroup of order n. We also show that the same bound is an upper bound for the number of conjugacy classes, provided that the k(G V)-conjecture holds for solvable G all primes dividing n.

The main result is that every torsion-free locally nilpotent group that is isomorphic to each of its nonnilpotent subgroups is nilpotent, that is, a torsion-free locally nilpotent group G that is not nilpotent has a non-nilpotent subgroup H that is not isomorphic to G.

The paper discusses modules over free nilpotent groups and demonstrates that faithful modules are more restricted than might appear at first glance. Some discussion is also made of applying the techniques more generally.

Bounds are obtained for the minimum number of generators for the fundamental groups of a family of closed 3-dimensional manifolds. A significant role has been played by the use of computers.

Let R be a ring with 1 and En (R) be the subgroup of GLn(R) generated by the matrices I + reij, r ∈ R, i ≠ j. We prove that the subgroup of consisting of the matrices of shape , where and , is (2, 3, 7)-generated whenever R is finitely generated and n, are large enough.

The Spelling Theorem of B. B. Newman states that for a one-relator group (a1, … | Wn), any nontrivial word which represents the identity must contain a (cyclic) subword of W±n longer than Wn−1. We provide a new proof of the Spelling Theorem using towers of 2-complexes. We also give a geometric classification of reduced disc diagrams in one-relator groups with torsion. Either the disc diagram has three 2-cells which lie almost entuirly along the bounday, or the disc diagram looks like a ladder. We use this ladder theorem to prove that a large class of one-relator groups with torsion are locally quasiconvex.

Let S be a subset of a group G such that S−1 = S. Denote by gr (S) the subgroup of G generated by S, and by ls(g) the length of an element g ∈ gr(S) relative to the set S. Suppose that V is a finite subset of a free group F of countable rank such that the verbal subgroup V (F) is a proper subgroup of F. For an arbitrary group G, denote by (G) the set of values in G of all the words from the set V. In the present paper, for amalgamated products G = A *HB such that A ≠ H and the number of double cosets of B by H is at least three, the infiniteness of the set {ls(g) | g ∈ gr(S)}, where S = (G) ∪ (G)−1, is estabilished.

Properties such as automaticity, growth and decidability are investigated for the class of finitely generated semigroups which have regular sets of unique normal forms. Knowledge obtained is then applied to the task of demonstrating that a class of semigroups derived from free inverse semigroups under certain closure operations is not automatic.

We show that if G is a finitely generated profinite group such that [x1, x2, …, xk] is Engel for any x1, x2, …, xk ∈ G, then γ(G) is locally nilpotent, and if [x1, x2, …, xk] has finite order for any x1, x2, …, xk ∈ G then, under some additional assumptions, γk(G) is locally finite.

Two projective nonsingular complex algebraic curves X and Y defined over the field R of real numbers can be isomorphic while their sets X(R) and Y(R) of R-rational points could be even non homeomorphic. This leads to the count of the number of real forms of a complex algebraic curve X, that is, those nonisomorphic real algebraic curves whose complexifications are isomorphic to X. In this paper we compute, as a function of genus, the maximum number of such real forms that a complex algebraic curve admits.

In a group G, um (G) denotes the subgroup of the elements which normalize every subnormal subgroup of G with defect at most m. The m-Wielandt series of G is then defined in a natural way. G is said to have finite m-Wielandt length if it coincides with a term of its m-Wielandt series. We investigate the structure of infinite groups with finite m-Wielandt length.

One of the converse statements to Lagrange's theorem is that, for each subgroup H of G and any prime factor p of |G: H|, there exists a subgroup K such that H≤K≤G with |K: H | = p. This paper treats integers n such that all groups of order n have this property.

A group G is locally graded if every finitely generated nontrivial subgroup of G has a nontrivial finite image. Let N (2, k)* denote the class of groups in which every infinite subset contains a pair of elements that generate a nilpotent subgroup of class at most k. We show that if G is a finitely generated locally graded N (2, k)*-group, then there is a positive integer c depending only on k such that G/Zc (G) is finite.