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?

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.

  • $\begingroup$ yes , i edited . thanks $\endgroup$ – user2993422 Sep 7 '15 at 16:53

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?

  • $\begingroup$ $A_3$ is of order 3 . and there are 4 groups in $A_4$ that are isomorphic to $A_3$ how do I proceed from here? thanks $\endgroup$ – user2993422 Sep 7 '15 at 16:58
  • $\begingroup$ $A_3$ is of order $3$, but $A_3$ has a more common name. Also, it doesn't matter which group of order $3$ that you use, as long as its image is all of $A_4/V$. $\endgroup$ – Michael Burr Sep 7 '15 at 18:10

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.