On exclusive semigroups.
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...
It is well-known that every bounded Abelian group is a direct sum of finite cyclic subgroups. We characterize those non-trivial bounded subgroups of an infinite Abelian group , for which there is an infinite subgroup of containing such that has a special decomposition into a direct sum which takes into account the properties of , and which induces a natural decomposition of into a direct sum of finite subgroups.
Suppose is a subgroup of the reduced abelian -group . The following two dual results are proved: If is countable and is an almost totally projective group, then is an almost totally projective group. If is countable and nice in such that is an almost totally projective group, then 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.