### A combinatorial problem in infinite groups.

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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...

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

The notion of free group is defined, a relatively wide collection of groups which enable infinite set summation (called commutative $\pi $-group), is introduced. Commutative $\pi $-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...

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

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 ${\gamma}_{n}\left(\u2329x\text{,}\phantom{\rule{4pt}{0ex}}{x}^{y}\u232a\right)$$={\gamma}_{n...}$