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On the cyclic subgroup separability of free products of two groups with amalgamated subgroup | | E. V. Sokolov
; | | Date: |
21 Aug 2007 | | Abstract: | Let $G$ be a free product of two groups with amalgamated subgroup, $pi$ be
either the set of all prime numbers or the one-element set {$p$} for some
prime number $p$. Denote by $Sigma$ the family of all cyclic subgroups of
group $G$, which are separable in the class of all finite $pi$-groups.
Obviously, cyclic subgroups of the free factors, which aren’t separable in
these factors by the family of all normal subgroups of finite $pi$-index of
group $G$, the subgroups conjugated with them and all subgroups, which aren’t
$pi^{prime}$-isolated, don’t belong to $Sigma$. Some sufficient conditions
are obtained for $Sigma$ to coincide with the family of all other
$pi^{prime}$-isolated cyclic subgroups of group $G$. It is proved, in
particular, that the residual $p$-finiteness of a free product with cyclic
amalgamation implies the $p$-separability of all $p^{prime}$-isolated cyclic
subgroups if the free factors are free or finitely generated residually
$p$-finite nilpotent groups. | | Source: | arXiv, 0708.2819 | | Services: | Forum | Review | PDF | Favorites | |
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