Let $G$ be a group with the property $G=G\_e\dot{\cup} G\_o$ with 
$G\_oG\_o\subseteq G\_e\leq G$.


($\dot{\cup}$ denotes disjoint union, $\leq$ is subgroup notation, and $G\_o^{-1}=\{x^{-1}: x\in G\_o\}$)


We observe that 


(1) $G\_o^{-1}=G\_o$, $G\_oG\_o=G\_e$, $G\_oG\_e=G\_eG\_o=G\_o$;


(2) $|G:G\_e|=2$, $|G|=2|G\_o|=2|G\_e|$;


(3) $(\mathbb{Z},+)$, $S\_n$ and $\mathbb{Z}\_{2m}$ enjoy the property.


Now, is there any characterization for such groups?



Even-odd partitioned groups!
Group $G$ has subsets $G\_e$ and $G\_o$ as mentioned above, iff $G$ has a normal subgroup of index 2. 
If $G\_e$ is a normal subgroup of index 2, then $G\_e$ and $G\_o=xG\_e$, $x\notin G\_e$, satisfied the condition of the problem. 


Now, suppose such subsets $G\_e$ and $G\_o$ exist. Since $G\_e$ is a subgroup, and $G\_e\cap G\_o=\emptyset$ $G\_o^{-1}\subseteq G\_o$, and so, $G\_o^{-1}=G\_o$. Therefore $G\_oG\_o=G\_e$. This implies that $G\_ex=xG\_e=G\_o$ for $x$ not in $G\_e$.This leads $G\_e$ to be a normal subgroup of index 2.


