1. ## show that...

Let H be subgroup of abelian group G, then show that:
i) $(aH)^-1=a^-1H$
ii) $(Ha)^2=Ha^2$

2. Originally Posted by GTK X Hunter
Let H be subgroup of abelian group G, then show that:
i) $(aH)^-1=a^-1H$
ii) $(Ha)^2=Ha^2$
Can you define $(aH)^-1$ and $(Ha)^2$ for me plz?

3. Originally Posted by GTK X Hunter
Let H be subgroup of abelian group G, then show that:
i) $(aH)^-1=a^-1H$
ii) $(Ha)^2=Ha^2$

Do you mean in the quotient group $G/H$?
This follows at once from the definitions of product in quotient groups. Read and understand well these definitions.

Tonio

4. take any $h\in H$
i) $(ah)^{-1}=h^{-1}a^{-1}=a^{-1}h^{-1}=a^{-1}h_o$
ii) $haha=hhaa=h_1a^2$
am i right? ^_^

5. Originally Posted by GTK X Hunter
take any $h\in H$
i) $(ah)^{-1}=h^{-1}a^{-1}=a^{-1}h^{-1}=a^{-1}h_o$
ii) $haha=hhaa=h_1a^2$
am i right? ^_^
It's simpler then that.

$\left(aH\right)\left(a^{-1}H\right)=(aa^{-1})H=H=e\implies a^{1}H=\left(aH\right)^{-1}$. Similarly for the second.