1
$\begingroup$

Is there two semidirect products of order $16$ with $C_2$ normal subgroup?

How many groups of the form $C_4 \times C_2$ : $C_2$ are there ?

Is it one expression for two groups ? or more?

$\endgroup$
  • $\begingroup$ You are looking for maps $C_4 \times C_2 \to \mathrm{aut}(C_2)$. So ask yourself what $\mathrm{aut}(C_2)$ is. $\endgroup$ – Myself Aug 24 '14 at 21:26
  • $\begingroup$ could you explain more ? what do you mean? $\endgroup$ – math math Aug 24 '14 at 21:27
  • $\begingroup$ You should probably add to your question the definition of semi-direct product you're most comfortable with if you don't understand my comment. $\endgroup$ – Myself Aug 24 '14 at 21:28
  • $\begingroup$ ok thanks for help $\endgroup$ – math math Aug 24 '14 at 21:29
2
$\begingroup$

A semi direct product $A\rtimes B$ is given by a group action of $B$ onto $A$, or equivalently by a group homomorphism $B\to \mathrm{Aut}(A)$, where $\mathrm{Aut}(A)$ is the group of automorphisms of $A$. Moreover, the semi direct product is isomorphic to $A\times B$ if the action is trivial.

In your case, the normal subgroup $A$ is of size $2$, so there are two elements, one is the identity and the other of order $2$. You cannot switch them, so $\mathrm{Aut}(A)$ is trivial. Hence, $A\rtimes B=A\times B$ for each group $B$, if $A$ is of size $2$.

If you try to do the converse, so take $A=C_4\times C_2$ and $B=C_2$, then you need to find an action of $B$ onto $A$.

$\endgroup$
1
$\begingroup$

There are two groups with structure description $(C_4\times C_2)\rtimes C_2$ whose Gap Ids are $(16,3)$ and $(16,13)$. You may see this for more detail.

$\endgroup$

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.