In my book, I have a theorem that says the following:
Let $G$ be a group. If $G_1,G_2$ are subgroups such that:
- $G_1, G_2 \lhd G$
- $G_1G_2 = G$
- $G_1 \cap G_2 =\{e_G\}$
Then $G \cong G_1 \times G_2$
Later, there is a remark that says that the converse of the theorem also holds. So, I suppose this means, that if we have that $G \cong G_1 \times G_2$ for subgroups $G_1,G_2$, then the three conditions listed above hold.
The 'proof' goes as follows:
If $G = G_1 \times G_2$, then $G = H_1H_2 $ with $H_1 = G_1 \times \{e_{G_2}\}$ and $H_2 = \{e_{G_1}\} \times G_2$. The groups $H_1,H_2$ are normal in $G$ and $H_1 \cap H_2 = \{e\} \quad \triangle$
Can someone explain this please? I really don't get how this proves anything. They don't even start from $G \cong G_1 \times G_2$? I will award the bounty to the person who can give me a detailled and rigorous explanation.
Best Answer
Definition. The cartesian product of two groups $G=G_1\times G_2$ is a group under the operation $(a,b)(c,d)=(ac,bd)$ together with these two maps: $$ \begin{equation} p_1\colon G\to G_1;~(x_1,x_2)p_1=x_1\\ p_2\colon G\to G_2;~(x_1,x_2)p_2=x_2 \end{equation} $$
(Theorem. $p_1$ and $p_2$ are group homomorphisms, and $\forall x_1\in G_1,\forall x_2\in G_2,\exists|x\in G$ such that $xp_1=x_1$ and $xp_2=x_2$. Moreover let $$ \begin{equation} i_1\colon G_1\to G;~x_1i_1=(x_1,e_{G_2})\\ i_2\colon G_2\to G;~x_2i_2=(e_{G_1},x_2) \end{equation} $$ it results that $$i_np_m=\begin{cases}I_{G_n}&\text{ if }n=m\\ O_{G_nG_m}&\text{ if }n\ne m \end{cases}$$ where $n,m\in\{1,2\}$ and $O_{XY}\colon X\to Y;~x\mapsto e_Y$ with $Y$ group. )
Definition. A direct product of two groups usually written again as $G_1\times G_2$ but that we will write as $A=(\{G_1, G_2\}, \{q_1,q_2\})$ is a group under the operation $(a,b)(c,d)=(ac,bd)$ together with these two homomorphisms: $$ \begin{equation} q_1\colon A\to G_1\\ q_2\colon A\to G_2 \end{equation} $$ for which $\exists$ homomorphisms $j_k\colon G_k\to G$ where $k=1,2$ such that $$j_nq_m=\begin{cases}I_{G_n}&\text{ if }n=m\\ O_{G_nG_m}&\text{ if }n\ne m \end{cases}$$ where $n,m\in\{1,2\}$
Definition. A direct-product group isomorphism is a group isomorphism $T:A\to B$, where $A=(\{G_1,G_2\},\{q_1,q_2\})$ and $B=(\{G_1,G_2\},\{r_1,r_2\})$ direct products of two groups $G_1$ and $G_2$, such that $$ \begin{equation} r_m=Tq_m \end{equation} $$ where $m=1,2$
(Theorem. The cartesian product is a direct product, that is, $G_1\times G_2=(\{G_1, G_2\}, \{p_1, p_2\})$, where $p_1$ and $p_2$ have been defined in the first of our definitions above. Moreover any direct product $A$ of two groups $G_1$ and $G_2$ is direct-product group isomorphic to the cartesian product group $G_1\times G_2$)
All that said, what you name 'proof' is not the proof of the converse of the main theorem but a lemma, probably made difficult to read because the letter $G$ has been reused with a different meaning w.r.t. that used in the main theorem. So first let's rewrite it avoiding such problem:
Lemma. The cartesian product $G_1 \times G_2$ is such that $G_1 \times G_2 = H_1H_2 $ with $H_1 = G_1 \times \{e_{G_2}\}$ and $H_2 = \{e_{G_1}\} \times G_2$. The groups $H_1,H_2$ are normal in $G_1 \times G_2$ and $H_1 \cap H_2 = \{e\} \quad \triangle$
Now by the main theorem and this lemma it follows:
Corollary. The cartesian product of two groups $G_1\times G_2$ is isomorphic to a direct product of $H_1$ and $H_2$ where $H_1 = G_1 \times \{e_{G_2}\}$ and $H_2 = \{e_{G_1}\} \times G_2$, and in particular $G_1\times G_2\cong H_1\times H_2$, that is $H_1\times H_2=(\{G_1,G_2\},\{q_1, q_2\})$ where $$ \begin{equation} q_1\colon H_1\times H_2\to G_1;~(x_1,0,0,x_2)\mapsto x_1\\ q_2\colon H_1\times H_2\to G_2;~(x_1,0,0,x_2)\mapsto x_2 \end{equation} $$ and $\exists$ an isomorphism $T\colon H_1\times H_2\to G_1\times G_2$ such that $q_n=Tp_n$, where $n=1,2$, indeed $$T\colon (x_1,0,0,x_2)\mapsto(x_1,x_2)$$
Moreover:
Proof of the converse of the main theorem. Let $G$ be a direct product of two subgroups of its, $G=(\{G_1, G_2\}, \{r_1, r_2\})$. Then $G$ is direct-product group isomorphic to the cartesian product of those subgroups $G\cong G_1\times G_2$. But then by the corollary it is also $G\cong H_1\times H_2$. That means that $\exists U\colon G\to H_1\times H_2$ direct-product group isomorphism such that $r_n=Uq_n$, where $n=1,2$. For the corollary $H_1\times H_2$ has two subgroups that satisfy the conditions of the main theorem, but then thanks to $U$, $G$ have two subgroups satisfying those conditions as well.