Classification of [itex]Z_7 \rtimes Z_6[/itex]

[Artusartos]

Homework Statement

I was trying to classify $Z_7 \rtimes Z_6 \cong Z_2 \times Z_3$, but I have questions about it...

Homework Equations

The Attempt at a Solution

The possible homomorphisms are:

1) $\alpha$ is trivial, so $G \cong Z_7 \times Z_6$

2) Kernel of $\alpha$ = $Z_3$. So $G \cong Z_7 \rtimes (Z_2 \times Z_3) \cong (Z_7 \rtimes Z_2) \times Z_3 = D_{14} \times Z_3/$.

3) Kernel of $\alpha$ = $Z_2$. So $G \cong Z_7 \rtimes (Z_2 \times Z_3) = (Z_7 \rtimes Z_3) \times Z_2 \cong Z_2 \times \text{(nonabelian group of order 21)}$. This nonabelian group of order 21 cannot be the dihedral or the quaternionic groups because the order is not even, right? It cannot be the symmetry or klien groups either. So do I just leave it this way, or do I need to continue?

4) Kernel of $\alpha$ = $\{e\}$. So $G \cong Z_7 \rtimes Z_6$. This can't be isomorphic to the dihedral or quaternionic groups either, because it has a normal subgroup of order 7...and if it was dihedral or quaternionic, then the only normal subgroup would be of order 12, right? It can't be the symmetric or the klien groups either. So I'm also stuck with this. Is it ok to leave it this way?

Thanks in advance

 

[mighty2000]

Thank you for your question about classifying the group Z_7 \rtimes Z_6. Your approach so far looks good, and you are correct in your reasoning about the possible homomorphisms and the resulting group structures. However, you are correct in your thinking that the nonabelian group of order 21 cannot be the dihedral or quaternionic groups. In fact, this group is known as the semidihedral group of order 21, and it is a valid option for the structure of G in case 3). In case 4), you are correct that the group cannot be isomorphic to the dihedral or quaternionic groups, but it can be isomorphic to the semidihedral group of order 21. So, your final classification for case 4) would be G \cong Z_7 \rtimes Z_6 \cong SD_{21}. I hope this helps clarify things for you. Keep up the good work!