Let $G$ be a group with the property $G=G_e\dot{\cup} G_O$ with $G_oG_o^{-1}\subseteq G_o^{-1}G_o=G_e\leq G$.

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

We observe that the integer numbers group, $S_n$ and $\mathbb{Z}_{2m}$ enjoy the property. Moreover, if $G$ has no any element of order $2$ then $|G:G_e|=2$, $|G|=2|G_o|=2|G_e|$.

Now, how are these groups characterized?. Any ideas?, references?, classes of such groups?

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?

Let $G$ be a group with the property $G=G_e\dot{\cup} G_O$ with $G_oG_o^{-1}\subseteq G_o^{-1}G_o=G_e\leq G$, and with no any element of order $2$ .

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

We observe that $|G:G_e|=2$, $|G|=2|G_o|=2|G_e|$, and the integer numbers group, $S_n$ and $\mathbb{Z}_{2m}$ enjoy the property.

Now, how are these groups characterized?. Any ideas?, references?, classes of such groups?

Let $G$ be a group with the property $G=G_e\dot{\cup} G_O$ with $G_oG_o^{-1}\subseteq G_o^{-1}G_o=G_e\leq G$.

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

We observe that the integer numbers group, $S_n$ and $\mathbb{Z}_{2m}$ enjoy the property. Moreover, if $G$ has no any element of order $2$ then $|G:G_e|=2$, $|G|=2|G_o|=2|G_e|$.

Now, how are these groups characterized?. Any ideas?, references?, classes of such groups?