TITLE: Subsets which are "never closed" under group operation
QUESTION [6 upvotes]: I am wondering if there are some specific properties of groups for which there exists nonempty subset, say $\exists X\ne\emptyset,X\subset G$, s. t. $\forall g,h\in X$, its product $gh$ does not belongs to $X$. I didn't find any references where such pairs $(G,X)$ will be discussed. Under what name (in the literature) should I look for such subsets, or group which possesses them? I am interested in study properties of such objects (under additional assumptions) and looking for some references. As an example it will be interesting to know when such pairs can exists, how large $X$ can be, how many such $X$s are possible for a given group $G$, if $(G,X)$ is such a pair, for which subgroup $H$ of $G$, $(H,X)$ also have this property etc.
I will be grateful for any suggestion.

REPLY [6 votes]: These are called sum-free sets and are a central topic in additive combinatorics where people study exactly the kind of questions that you mention. 
See for example the paper Sum-free sets in abelian groups by Ben Green and Imre Ruzsa.