Let p and t are the cycles (123) and (12) of the symmetric group $\displaystyle S_3$.

a/ Proof that <p> is a normal subgroup of $\displaystyle S_3$, but <t> is not a normal subgroup in $\displaystyle S_3$

b/ Proof that the set H = {$\displaystyle p^k t^s$ |k=0,1,2;s=0,1;} is a subgroup of $\displaystyle S_3$

c/ Proof that H = $\displaystyle S_3$

I will appreciate any help. Thank you :)