# Group and subgroup

• Apr 5th 2010, 05:15 PM
bhitroofen01
Group and subgroup
(G;x) is a group ( e it's identity element). $H_1$ and $H_2$ are a subgroups from (G,x).
$H_1H_2$={ $x_1x_2$/ $(x_1;x_2$} $\in$ $H_1$x $H_2$}

1)-I must show that : $H_1H_2$ is a subgroup from(G;x) $\Longrightarrow H_1H_2=H_2H_1$
2)- We assume that $H_1$ and $H_2$ are finished and $H_1\cap H_2$={e} and:
$\phi$ : $H_1$x $H_2$ $\longrightarrow$ $H_1$ $H_2$
( $x_1$; $x_2$) $\longrightarrow$ $x_1$ $x_2$

I must show that $\phi$ is surjective and Card( $H_1H_2$)=Card( $H_1$)Card( $H_2$)
I don't know anything!!!!!!! Can you give me some help please????
• Apr 5th 2010, 05:20 PM
Drexel28
Quote:

Originally Posted by bhitroofen01
Let $H_1,H_2\leqslant G$ be such that $H_1\cap H_2=\{e\}$. Define $\phi:H_1\times H_2\to H_1H_2$ by $(h_1,h_2)\overset{\phi}{\longmapsto}h_1h_2$. Prove that $\phi$ is surjective.

Is the above the second part?
• Apr 5th 2010, 05:23 PM
bhitroofen01
yes
• Apr 5th 2010, 05:25 PM
Drexel28
Quote:

Originally Posted by bhitroofen01
yes

Let's see some work.
• Apr 5th 2010, 05:31 PM
bhitroofen01
$H_1$ $H_2$ is a subgroup from (G;x) $\Longleftrightarrow$ $H_1$x( $H_2)^{-1}\in H_1H_2$
• Apr 5th 2010, 09:31 PM
FancyMouse
Did you drop any assumptions? The statement doesn't seem true to me.