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For a finitely generated group, we study the relations between its rank, the maximal rank of its free quotient, called co-rank (inner rank, cut number), and the maximal rank of its free abelian quotient, called the Betti number. We show that any combination of the group's rank, co-rank, and Betti number within obvious constraints is realized for some finitely presented group (for Betti number equal to rank, the group can be chosen torsion-free). In addition, we show that the Betti number is additive...

Corrigendum to: "Aspherical four-manifolds and the centres of two-knot groups".

Jonathan A. Hillman (1983)

Commentarii mathematici Helvetici

Die maximale lokale Erklärung einer gesättigten Formation

Klaus Doerk (1973)

Mathematische Zeitschrift

Endliche minkowskische Ebenen von Charakteristik ... 2.

Rolf Lingenberg (1975)

Journal für die reine und angewandte Mathematik

Extensions of Certain Homomorphisms of Subsemi-groups to Homomorphisms of Groups

D.Z. DJOKOVIC, J.A. BAKER, J. ACZÉL (1971)

Aequationes mathematicae

Extensions of homomorphisms of subsemigroups of a group.

K.E. Osondu (1977/1978)

Semigroup forum

Groups whose proper subgroups are locally finite-by-nilpotent

Amel Dilmi (2007)

Annales mathématiques Blaise Pascal

If $𝒳$ is a class of groups, then a group $G$ is said to be minimal non $𝒳$ -group if all its proper subgroups are in the class $𝒳$ , but $G$ itself is not an $𝒳$ -group. The main result of this note is that if $c > 0$ is an integer and if $G$ is a minimal non $(ℒℱ) 𝒩$ (respectively, $(ℒℱ) 𝒩_{c}$ )-group, then $G$ is a finitely generated perfect group which has no non-trivial finite factor and such that $G / F r a t (G)$ is an infinite simple group; where $𝒩$ (respectively, $𝒩_{c}$ , $ℒℱ$ ) denotes the class of nilpotent (respectively, nilpotent of class at most $c$ , locally...

Homology theory in the alternative set theory I. Algebraic preliminaries

Jaroslav Guričan (1991)

Commentationes Mathematicae Universitatis Carolinae

The notion of free group is defined, a relatively wide collection of groups which enable infinite set summation (called commutative $π$ -group), is introduced. Commutative $π$ -groups are studied from the set-theoretical point of view and from the point of view of free groups. Commutativity of the operator which is a special kind of inverse limit and factorization, is proved. Tensor product is defined, commutativity of direct product (also a free group construction and tensor product) with the special...

Homology theory in the AST III. Comparison with homology theories of Čech and Vietoris

Jaroslav Guričan (1993)

Commentationes Mathematicae Universitatis Carolinae

The isomorphism between our homology functor and these of Vietoris and Čech is proved. Introductory result on dimension is proved.

Hyper–(Abelian–by–finite) groups with many subgroups of finite depth

Fares Gherbi, Tarek Rouabhi (2007)

Annales mathématiques Blaise Pascal

The main result of this note is that a finitely generated hyper-(Abelian-by-finite) group $G$ is finite-by-nilpotent if and only if every infinite subset contains two distinct elements $x$ , $y$ such that $γ_{n} (〈x, x^{y}〉)$ $= γ_{n ...}$

Inert actions on periodic points.

Kim, K.H., Roush, F.W., Wagoner, J.B. (1997)

Electronic Research Announcements of the American Mathematical Society [electronic only]

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