Can the alternating groups $A_4$ and/or $A_5$ be written as a semidirect product?

Question

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

H,N

is that correct?

for A4

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.

Answer 1

You are correct that since $A_5$ has no nontrivial normal subgroups, it can't be written as a nontrivial semidirect product.

Now, for $A_4$, the copy of $V=(\mathbb{Z}/2)^2$ within the group is a nontrivial normal subgroup, so it is natural to try to use this. Now, you must consider $A_4/V$. By considering the order, this is a group of order $3$. How many subgroups are there of order 3? Finally to write $A_4$ as a semi-direct product, you need a copy of that quotient group as a subgroup of $A_4$. Can you find such a subgroup?