I need to tell if
the alternating groups $A_4$ and \ or $A_5$ can be written as a semidirect product of non trivial subgroups.
what I tried:
I think there is a theorem that says that it is enough to show two subgroups
is that correct?
I know that the four Klein group $V_4$ is normal to $A_4$
and that $V_4$ has no common elements with $A_3$
i assume $A_3 * V_4 = A_4$ (is that correct) ?
so i think that i can write $A_4$ as a semiproduct of $V_4$ and $A_3$
not sure about it..
about $A_5$ I read that for n>=5 An doesn't have non trivial normal subgroups.
so because of that it can't be written as a semidirect product?
any help will be appreciated.