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On exponential growth rates for free groups.

Malik Koubi (1998)

Publicacions Matemàtiques

Let Fp be a free group of rank p ≥ 2. It is well-known that, with respect to a p-element generating set, that is, a basis, the exponential growth rate of Fp is 2p-1. We show that the exponential growth rate τ of a group G with respect to a p-element generating set X is 2p-1 if and only if G is free on X; otherwise τ < 2p-1. We also prove that, for any finite generating set X of Fp which is disjoint from X-1, the exponential growth rate τ of Fp with respect to X is 2p-1 if and only if X is...

On extensions of bounded subgroups in Abelian groups

S. S. Gabriyelyan (2014)

Commentationes Mathematicae Universitatis Carolinae

It is well-known that every bounded Abelian group is a direct sum of finite cyclic subgroups. We characterize those non-trivial bounded subgroups H of an infinite Abelian group G , for which there is an infinite subgroup G 0 of G containing H such that G 0 has a special decomposition into a direct sum which takes into account the properties of G , and which induces a natural decomposition of H into a direct sum of finite subgroups.

On extensions of primary almost totally projective abelian groups

Peter Vassilev Danchev (2008)

Mathematica Bohemica

Suppose G is a subgroup of the reduced abelian p -group A . The following two dual results are proved: ( * ) If A / G is countable and G is an almost totally projective group, then A is an almost totally projective group. ( * * ) If G is countable and nice in A such that A / G is an almost totally projective group, then A is an almost totally projective group. These results somewhat strengthen theorems due to Wallace (J. Algebra, 1971) and Hill (Comment. Math. Univ. Carol., 1995), respectively.

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