Let be the given trigonometric series, then the formal product of them is defined by where the last series is supposed to be convergent for every n.

Many one-relator groups with center have been shown to be of the form A necessary and a sufficient condition for the sequence (P1, Q1, P2, Q2,…,Pt, Qt) are given in order for groups of the above form to be one-relator groups.

Let K be a skewfield, E a left vector space over K, r an integer ≧ 1 and Gr(E) the set of all r-dimensional subspaces of E, called the Grassmannian of index r. The function d(A, B) = r — dim (A∩B) is a distance on Gr(E). If K′ is a skewfield and E′ a left vector space over K1, then any semilinear ismorphism u: E → E1(relative to an isomorphism K → K') induces a distance preserving bijection Gr(u):Gr(E) → Gr(E′). When E has finite dimension n and 2r = n, another example of such a mapping is obtained by taking K′ = Kop, E′ = E* and defining wr: Gr(E) → Gr(E*) to be wr(A) = {f ∈ E*&f(A) = 0}.

In [6] and [2] Markov and Graev introduced their respective concepts of a free topological group. Graev's concept is more general in the sense that every Markov free topological group is a Graev free topological group. In fact, if FG(X) is the Graev free topological group on a topological space X, then it is the Markov free topological group FM(Y) on some space Y if and only if X is disconnected. This, however, does not say how FG(X) and FM(X) are related.

A class of groups forms a (subnormal) coalition class, or is (subnormally) coalescent, if whenever H and K are subnormal -subgroups of a group G then their join <H, K> is also a subnormal -subgroup of G. Among the known coalition classes are those of finite groups and polycylic groups (Wielandt [15]); groups with maximal condition for subgroups (Baer [1]); finitely generated nilpotent groups (Baer [2]); groups with maximal or minimal condition on subnormal subgroups (Robinson [8], Roseblade [11, 12]); minimax groups (Roseblade, unpublished); and any subjunctive class of finitely generated groups (Roseblade and Stonehewer [13]).

The study of minimal irregular p-groups was initiated by Mann [9]. He defines such a group to be a finite irregular p-group in which every proper section is regular. He then deduces a large number of properties of such groups and shows, by construction, that there is no bound on their exponent. In this note methods involving varieties of groups are used to show that all minimal irregular p-groups can be ‘built up’ from cyclic groups, groups of exponent p and a relatively small class of metabelian minimal irregular p-groups. To phrase this more precisely, say that a group P is derivable from a family of groups {Qi} if P is a homomorphic image of a subdirect product of the Qi. Then we will prove as Theorem 2.2: a minimal irregular p-group of exponent pn and nilpotency class c is derivable from a family of three groups: i) a cyclic group of order pn, ii) a 2-generator monolithic group of exponent p and class c, iii) a minimal irregular p-group whi h is metabelian, has class p and has exponent p2.

The four colour conjecture is well known to be equivalent to the proposition that every trivalent planar graph without an isthmus (i.e. an edge whose removal disconnects the graph) has an edge colouring in three colours ([1], p. 121). By an edge colouring we mean an assignment of colours to the edges of the graph so that no two edges of the same colour meet at a common vertex, and the graph is n-valent if n edges meet at each vertex. An edge colouring by three colours is called a Tait colouring; a trivalent graph which has a Tait colouring can be split in three edge-disjoint 1-factors, i.e. spanning monovalent subgraphs.

A subgroup H of a group G is said to be a permutablesubgroup of G if HK = KH 〈H, K〉 for all subgroups K of G. It is known that a core-free permutable subgroup H of a finite group G is always nilpotent [5]; and even when G is not finite, H is always a subdirect product of finite nilpotent groups [11]. Thus nilpotency is a measure of the extent to which a permutable subgroup differs from being normal. Examples of non-abelian, core-free, permutable subgroups are rare and difficult to construct. The first, due to Thompson [12], had class 2. Further examples of class 2 appeared in [8]. More recently Bradway, Gross and Scott [1] have constructed corresponding to each positive integer c and each prime p ≥ c, a finite p-group possessing a core-free permutable subgroup of class c. In [3] Gross succeeded in dispensing with the requirement p ≥ c.

Let be a non-cyclic free presentation of a group G, the lower central series of R. Then Rn/Rn+1, n ≧ 1, can be regarded as a right G-module by defining the action of G via conjugation in F. We wish to investigate the annihilators of these modules which we call higher relation modules.

In 1961 Graham Higman [1] proved that a finitely generated group is a subgroup of a finitely presented group if, and only if, it is recursively presented. Therefore a finitely generated metabelian group can be embedded in a finitely presented group.

Let Sn denote the sphere of all points in Euclidean space Rn + 1 at a distance of 1 from the origin and Dn + 1 the ball of all points in Rn + 1 at a distance not exceeding 1 from the origin The space X is said to be aspherical if for every n ≧ 2 and every continuous mapping: f: Sn → X, there exists a continuous mapping g: Dn + 1 → X with restriction to the subspace Sn equal to f. Thus, the only homotopy group of X which might be non-zero is the fundamental group τ1(X, *) ≅ G. If X is also a cell-complex, it is called a K(G, 1). If X and Y are K(G, l)'s, then they have the same homotopy type, and consequently

One of the best known classes of non-Desarguesian planes is the calss of Hall planes (see Hall [2]). In [6] Hanna Neumann showed that the finite Hall planes of old order possess subplanes of order two (i.e., Fano subplanes)1. Kirkpatrick [5] has considered a type of plane which is generalization of the Hall planes and which he calls generalized Hall planes. In this paper we will give a sufficient condition that a finite generalized Hall plane possesses Fano subplanes. Some examples of odd order planes to which the condition applies shall be exhibited.

A right module A over a ring R is called finitely injective if every diagram of right R-modules of the form where X is finitely generated and the row is exact, can be imbedded in a commutative diagram The finitely injective modules turn out to have the nice property of being ‘strongly pure’ in all the containing modules and, in particular, are absolutely pure in the sense of Maddox [8]. For this reason we call them strongly absolutely pure (for short, SAP) modules. Several other characterizations of the SAP modules are obtained. It is shown that every module admits a ‘SAP hull’. The concepts of finitely M-injective and finitely quasi-injective modules are then investigated. A subclass of finitely quasi-injective modules, called the strongly regular modules, is studied in some detail and it is shown that, if R/J is right Noetherian, then the strongly regular right R-modules coincide with the semi-simple R-modules.

The theory of alternating bilinear forms on finite dimensional vector spaces V is well understood; two forms on V are equivalent if and only if they have equal ranks. The situation for alternating trilinear forms is much harder. This is partly because the number of forms of a given dimension is not independent of the underlying field and so there is no useful canonical description of an alternating trilinear form.

Under certain general conditions an n x n stochastic matrix P = (pij), where

is known to possess the “ergodic” property. This means that there exists a finite matrix E such that

In his work [5] on subgroups of one relator groups, Moldavanski observed that if G is a one relator group whose defining relator R is cyclically reduced and has exponent sum zero on some generator occurring in it, then G is an HNN extension of a one relator group H whose defining relator is shorter than R. This observation, together with Britton's Lemma, can be used to give rather easy proofs of the basic results on one relator groups. To exposit this point of view, we give here a proof of the Freiheitssatz, the solvability of the word problem for one relator groups, and the theorem classifying elements of finite order in one relator groups. In particular, the solution obtained for the word problem is often easy to apply. We also give a proof of the “Spelling Theorem” of Newman [6].