Let $G$ be an abelian group, $H\subset G$ a subgroup such that if $nx\in H$ for $n\in \bf{Z}$ and $x\in G$ then $x\in H$. Is there a name for subgroups of abelian groups $H$ with this property?
Asked
Active
Viewed 31 times
2
-
1Can you be more specific about your quantifiers? Do you mean 'if there's any $n$ such that $nx\in H$ then $x\in H$'? As things stand it reads like 'if for all $n$ and $x$, $nx\in H$, then $x\in H$', which is trivially true (just take $n=1$...) – Steven Stadnicki May 28 '20 at 16:26
-
Yeah, changed. The other meaning would always be true of course :P. – Joshua Tilley May 28 '20 at 16:28
-
1I don't know of a name, but a good guess might be "radical subgroup", inspired by the radical ideals of a ring. – HallaSurvivor May 28 '20 at 16:30
-
1For a related (but different) notion see divisible group. – Dietrich Burde May 28 '20 at 16:51
-
I found this https://ncatlab.org/nlab/show/divisible+group – Joshua Tilley Nov 14 '20 at 12:43