Classification of groups of order $18$

Question

I was just going through the first classification-

$$|G|=18=3^2\times 2$$

Then $G$ has a subgroup of order $9$(normal, say $K$) and a subgroup of order $2$ (say $H$).

I want someone to help me with the first case.

Let $$\theta:H \to{\rm Aut}(K)$$

$$\theta :h \mapsto \theta_h$$

Assume that $H \cong \mathbb{Z_3} \times \mathbb{Z_3}$. Now ${\rm Aut}(\mathbb{Z_3} \times \mathbb{Z_3}) \cong GL_2(\mathbb{Z}_3)$.

If $\theta$ is the identity map then $G$ is isomorphic to $H \times K$.

Otherwise $\theta(1)$ must be of order $2$. How do I determine the isomorphism and proceed after this.

Answer 1

Conjugate elements of order $2$ in ${\rm GL}_2(3)$ result in isomorphic semidirect products, so you need to find representatives of the conjugacy classes of elements of order $2$ in ${\rm GL}_2(3)$.

That is not difficult, because all such elements are similar to a diagonal matrix, whose entries must be $1$ or $-1$. So there are two such classes, giving rise to two nonabelian groups of order $18$ with normal subgroup $C_3^2$.

Answer 2

According to your notations, $$\theta:H\longrightarrow Aut(K),\ \text{such that}\ h\longmapsto \theta_h\ \text{where,}\ \theta_h(k)=hkh^{-1},\ \forall h\in H\ \&\ \forall k\in K. $$

Case $1$ : When $H\cong \mathbb{Z}_2$ and $K\cong \mathbb{Z}_9$, then $\theta:\mathbb{Z}_2\longrightarrow Aut(\mathbb{Z}_9)$. From here we will get $\theta_0$ and $\theta_1$. $$\theta_0(a)=a\ \text{and} \ \theta_1(b)=(b+2)(mod\ 9),\ \forall a,b\in \mathbb{Z}_9.$$

Case $2$ : $H\cong \mathbb{Z}_2\ \&\ K\cong \mathbb{Z}_3\times\mathbb{Z}_3 $. So, we can wrrite $$H=\{e,h_1\} \ \text{and}\ K=\{e,\ k_1,\ k_2,\ k_1^2,\ k_2^2,\ k_1k_2,\ k_1^2k_2,\ k_1k_2^2,\ k_1^2k_2^2 \}\cdots\cdots\cdots(1)$$ where, $o(h_1)=2$, $o(k_1)=o(k_2)=3$ and $k_1,\ k_2$ commute with each other. The identities are same $(e_H=e_K=e)$, because both are the subgroups of $G$. So, all the elements of $H\ \&\ K$ belongs to $G$. $\theta':H\longrightarrow Aut(K)$. From here we will get $\theta'_e$ and $\theta'_{h_1}$. $$\theta'_e(k)=eke^{-1}=k\ \text{and}\ \theta'_{h_1}(k)=h_1kh_1^{-1}=h_1kh_1,\ \forall k\in K.$$

From both the cases we have the following possible structures, $$G\cong \mathbb{Z}_9\rtimes_{\theta_0}\mathbb{Z}_2,\ G\cong \mathbb{Z}_9\rtimes_{\theta_1}\mathbb{Z}_2,\ G\cong K\rtimes_{\theta'_e}H,\ G\cong K\rtimes_{\theta'_{h_1}}H$$ where, $H\ \&\ K$ follows $(1)$