Showing papers in "Journal of The Australian Mathematical Society in 1973"
TL;DR: In this paper, it was shown that a group G is a finite extension of a free group if and only if G is an HNN group where K is a tree product of a finite number of finite groups (the vertices of K), and each (associated) subgroup Li, Mi is a subgroup of a vertex of K.
Abstract: Using Stalling's characterization [11] of finitely generated (f. g.) groups with infinitely many ends, and subgroup theorems for generalized free products and HNN groups (see [9], [5], and [7]), we give (in Theorem 1) a n.a.s.c. for a f.g. group to be a finite extension of a free group. Specifically (using the terminology extension of and notation of [5]), a f.g. group G is a finite extension of a free group if and only if G is an HNN group where K is a tree product of a finite number of finite groups (the vertices of K), and each (associated) subgroup Li, Mi is a subgroup of a vertex of K.
174 citations
TL;DR: In this article, a partial solution to the problem of characterizing the rings R for which Abelian groups can be characterized is presented, which is the same problem posed by Fuchs in his well-known book "Abelian Groups".
Abstract: One of the still unsolved problems posed by Fuchs in his well-known book “Abelian Groups” [2] is Problem 45: characterize the rings R for which . I present here a partial solution.
78 citations
66 citations
TL;DR: The LEmma theorem states that a group G is SQ-universal if every countable group H is embeddable in some factor group of G as mentioned in this paper, and if H is a subgroup of finite index in a group H, then H is SQuniversal.
Abstract: Following a suggestion of G. Higman we say that the group G is SQ-universal if every countable group is embeddable in some factor group of G. It is a well-known theorem of G. Higman, B. H. Neumann and Hanna Neumann that the free group of rank 2 is sq-universal in this sense. Several different proofs are now available (see, for example, [1] or [9]). It is my intention to prove the LEmma. If H is a subgroup of finite index in a group G, then G is SQ-universal if and only if H is SQ-universal.
61 citations
TL;DR: In this article, a criterion for a sequence of ordinals to be generic over a transitive model of ZFC with respect to a notation of forcing first considered by Prikry in his Doctoral dissertation was established.
Abstract: I establish here a criterion for a sequence of ordinals to be generic over a transitive model of ZFC with respect to a notation of forcing first considered by Prikry in his Doctoral dissertation [2]. In Section 0 I review some notation, in Section 1 I list some facts about measurable cardinals, and in Section 2, after giving Prikry's result, I state and prove mine. Theorem 2.2 was proved during my brief stay at Monash University in Melbourne in June 1969. I thank Professor Crossley of that organisation for his hospitality. The paper was written in my sister's house in Pakistan.
58 citations
TL;DR: In this article, it was shown that for every n ≧ 2 and every continuous mapping f : S n → X, there exists a continuous mapping g : D n + 1 → X with restriction to the subspace S n equal to f.
Abstract: Let S n denote the sphere of all points in Euclidean space R n + 1 at a distance of 1 from the origin and D n + 1 the ball of all points in R n + 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 : S n → X , there exists a continuous mapping g : D n + 1 → X with restriction to the subspace S n equal to f . Thus, the only homotopy group of X which might be non-zero is the fundamental group τ 1 ( 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
53 citations
TL;DR: In this paper, it was shown that the free centre-by-metabelian group of rank n has a finite elementary abelian subgroup Hn of rank 4 of the subgroup contained in the centre of the group and is isomorphic to a group of 3 x 3 matrices over a finitely generated integral domain of characteristic zero.
Abstract: Let be the free centre-by-metabelian group of rank n. In this paper, our main result is the followingTheorem. For n ≧ 4, Gn has a finite elementary abelian subgroup Hn of rank . More precisely, Hn is a minimal fully invariant subgroup contained in the centre of Gn and Gn/Hn is isomorphic to a group of 3 x 3 matrices over a finitely generated integral domain of characteristic zero.
51 citations
45 citations
TL;DR: In this paper, it was shown that strongly pure modules have the nice property of being strongly pure in all the containing modules and are called strongly absolutely pure (for short, SAP) modules.
Abstract: 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.
44 citations
TL;DR: The lattice of annulets as mentioned in this paper is a sublattice of the Boolean algebra of all annihilator ideals in a distributive lattice L with 0, which is the dual of the so-called filets (carriers).
Abstract: In a distributive lattice L with 0 the set of all ideals of the form (x]* can be made into a lattice A0(L) called the lattice of annulets of L. A 0(L) is a sublattice of the Boolean algebra of all annihilator ideals in L. While the lattice of annulets is no more than the dual of the so-called lattice of filets (carriers) as studied in the theory of l-groups and abstractly for distributive lattices in [1, section4] it is a useful notion in its own right. For example, from the basic theorem of [3] it follows that A 0(L) is a sublattice of the lattice of all ideals of L if and only if each prime ideal in L contains a unique minimal prime ideal.
42 citations
TL;DR: In this paper, the authors use standard graph notation and definitions, as in [1], in particular Kn is the complete graph on n vertices and Kn, n is the regular complete bigraph of order 2n.
Abstract: We use standard graph notation and definitions, as in [1]: in particular Kn is the complete graph on n vertices and Kn, n is the regular complete bigraph of order 2n.
TL;DR: In this paper, the authors consider various subspaces of a semitopological semigroup and determine what topological algebraic structure can be introduced into the spaces of means on the sub-spaces and into the spectra of the C*-sub-algebras of C(S) they generate.
Abstract: Suppose S is a semitopological semigroup. We consider various subspaces of C(S) and determine what topological algebraic structure can be introduced into the spaces of means on the subspaces and into the spectra of the C*-sub-algebras of C(S) they generate.
TL;DR: In this article, the Freiheitssatz, the solvability of the word problem for one relator groups, and the theorem classifying elements of finite order in one-relator groups are presented.
Abstract: 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].
TL;DR: In this paper, the authors consider a finite 2-group having a minimal generating set { x 1, …, x r } and show that r = d (G ) is an invariant of G.
Abstract: Let G be a finite 2-group having a minimal generating set { x 1 , …, x r } so that r = d ( G ) is an invariant of G . Suppose further that G has a presentation then .
TL;DR: In this article, it was shown that a regular semigroup is always a regular subsemigroup of S and investigated relationships between it and S, where = S is of particular interest.
Abstract: Suppose S is a regular semigroup and E is its set of idempotents. If E is subsemigroup of S, then S has been called orthodox and studied recently by Hall [3], Meakin [6], and Yamada [8]. In this paper we assume that E is not (necessarily) a subsemigroup of S and consider the subsemigroup generated by E, denoted . If E denotes the set of all elements of S which can be written E, denoted . If E denotes the set of all elements of S which can be written as the product of n (not necessarily distinct) idempotents of S, then . We show that is always a regular subsemigroup of S and investigate relationships between it and S. The case where = S is of particular interest to us; such semigroups will be referred to as idempotent-generated regular semi- groups.
TL;DR: In this article, the authors introduced the concept of Markov and Graev free topological groups (FG(X)) and showed that FG(X) is the Graev-free topological group on a topological space X, then it is the Markov free group FM(Y) on some space Y if and only if X is disconnected.
Abstract: 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.
TL;DR: In this paper, several interpretations of the inequality for all such inequalities were given and the following was proved: the following interpretation was proved in [2,5,6,7] a.o.
Abstract: 1. In [2,5,6,7] a.o. several interpretations of the inequality for all such that were given and the following was proved.
TL;DR: The homology functor from the category of free abelian chain complexes and homotopy classes of maps to that of graded groups is full and replete (surjective on objects up to isomorphism) and reflects isomorphisms as mentioned in this paper.
Abstract: The homology functor from the category of free abelian chain complexes and homotopy classes of maps to that of graded abelian groups is full and replete (surjective on objects up to isomorphism) and reflects isomorphisms. Thus such a complex is determined to within homotopy equivalence (although not a unique homotopy equivalence) by its homology. The homotopy classes of maps between two such complexes should therefore be expressible in terms of the homology groups, and such an expression is in fact provided by the Kunneth formula for Hom, sometimes called ‘the homotopy classification theorem’.
TL;DR: In this paper, it was shown that p is a bigenetic property of p-groups (or more simply, p is bigENetic in p-group) whenever all two-generator subgroups of G have p.
Abstract: Let be a class and p a property of groups. We say that p is a bigenetic property of p-groups (or more simply, p is bigenetic in p-groups) if an p-group G has the property p whenever all two-generator subgroups of G have p.
TL;DR: In this paper, it was shown that if F is a finitely generated free associative algebra over the field and R is an ideal of F, then F/R2 is finitely presented if and only if F/ R has finite dimension.
Abstract: We prove here that if F is a finitely generated free associative algebra over the field and R is an ideal of F, then F/R2 is finitely presented if and only if F/R has finite dimension. Amitsur, [1, p. 136] asked whether a finitely generated algebra which is embeddable in matrices over a commutative f algebra is necessarily finitely presented. Let R = F′, the commutator ideal of F, then [4, theorem 6], F/F′2 is embeddable and thus provides a negative answer to his question. Another such example can be found in Small [6]. We also show that there are uncountably many two generator I algebras which satisfy a polynomial identity yet are not embeddable in any algebra of n xn matrices over a commutative algebra.
TL;DR: In this paper, the authors studied abelian /-groups on which a tight Riesz order can be denned, such that = is precisely the associated order, and gave certain sufficient conditions for an /-group to have a CTRO, a useful necessary and sufficient condition, and some negative results concerning necessary conditions.
Abstract: Loy and Miller [4] have studied tight Riesz groups (G, g ) , without pseudozeros, and with (G, = )̂ an /-group. In this paper we study abelian /-groups (G, = )̂ on which a tight Riesz order can be denned, such that =̂ is precisely the associated order. Such an order we call a CTRO (compatible tight Riesz order). We give certain sufficient conditions for an /-group to have a CTRO, a useful necessary and sufficient condition, and some negative results concerning necessary conditions. The class of CTRO of a given /-group, ordered by set inclusion of positive cones, is directed downwards, has a maximal element, but usually is not directed upwards and has no smallest or greatest elements.
TL;DR: In this article, the authors studied the problem of the representation of near-rings, and in particular of a faithful representation, which is equivalent to the adjoining of an identity, and gave some special conditions under which a near-ring with two-sided zero exists.
Abstract: In this paper we study the problem of the representation of d.g. near-rings, and in particular the problem of a faithful representation, which is equivalent to the adjoining of an identity. This problem has been considered by Malone [5] and Malone and Heatherly [6] and [7]. They have shown that a finite near-ring with two sided zero can be embedded in the d.g. near-ring generated by the inner automorphisms of a suitable group, and that an identity can always be adjoined to a near-ring with two sided zero. They have also given some special conditions under which a faithful representation of a d.g. near-ring exists.
TL;DR: In this paper, the authors give an alternative description of free inverse semigroups and show how to obtain the canonical form for the elements of a free inverse semiigroup from Scheiblich's construction.
Abstract: In an important recent paper H. E. Scheiblich gave a construction of free inverse semigroups that throws considerable light on their structure [1]. In this note we give an alternative description of free inverse semigroups. What Scheiblich did was to construct a free inverse semigroup as a semigroup of isomorphisms between principal ideals of a semilattice E , say, thus realising free inverse semigroups as inversee subsemigroupss of the semigroup T E , a kind of inverse semigroup introduced and exploited by W. D. Munn [2]. We go instead directly to canonical forms for the elements of a free inverse semigroup. The connexion between our construction and that of Scheiblich's will be clear. There are several alternative procedures possible to reach our construction on which we comment on the way.
TL;DR: In this paper, the authors present proofs that are simple and possibly new for two theorems that appear in Rankin's book [6] and prove that they are not new.
Abstract: We supply proofs that are simple, and possibly partly new, for two theorems that appear in Rankin's book [6].
TL;DR: In this paper, it was shown that the class of finite soluble groups is universal and a Schunck class is one which satisfies: (a) = Q, and (b) contains all groups G such that Q (G ) ∩ B ⊆.
Abstract: In his Canberra lectures on finite soluble groups, [3], Gaschutz observed that a Schunck class (sometimes called a saturated homomorph) is { Q , E φ , D 0 }-closed but not necessarily R 0 closed(*). In Problem 7·8 of the notes he then asks whether every { Q , E φ , D 0 }-closed class is a Schunck class. We show below with an example † that this is not the case, and then we construct a closure operation R 0 satisfying D o o R o such that is a Schunck class if and only if = { Q E φ, R o }. In what follows the class of finite soluble groups is universal. Let B denote the class of primitive groups. We recall that a Schunck class is one which satisfies: (a) = Q , and (b) contains all groups G such that Q ( G ) ∩ B ⊆.