Adjoining elements to group

Question

I finally got some interesting class of examples in which deleting countable element or finite element from a group will remain with group structure.(structure with associated with four properties say, closure,associative,existance of identity and inverses with Binary operations *) But,the question raises about the other side of this above notion. i.e. after adjoining countable or finite element to a group then the new structure of group will remain group. any suggestions regarding this is highly appreciated.

Answer 1

I'm not really sure what you mean, but obviously there are many many examples. First the first part. Consider any group $G$ that has a strict subgroup $H$, then deleting $G\setminus H$ from $G$ leaves a group. There are many examples in which $G\setminus H$ is countably infinite or finite. Now conversely, suppose that $G$ is a group and deleting $U$ from $G$ leaves a group. Then $G\setminus U$ is a subgroup of $G$. Thus the first part of your question is simply looking for subgroups. (after deleting only finite or countably infinite elements).

For the second part. Suppose that $G$ and $H$ are groups. Then $G\oplus H$ is another group strictly containing $G$. If $H$ is finite or countably infinite, then we adjoined only finitely many or countably infinitely many elements to $G$ such that the result is again a group.

So unless I don't understand what you are asking, I'd say that this is a pretty meaningless question.