Groups of order 18

Let \(G\) be a group of order \(18 = 2 \cdot 3^2\). By Sylow’s theorem:

• The number \(n_3\) of Sylow 3-subgroups satisfies \(n_3 \equiv 1 \pmod 3\) and \(n_3 \mid 2\), so \(n_3 = 1\).

• Let \(P\) be the unique Sylow 3-subgroup of \(G\). Then \(P \normal G\), and \(\abs P = 9\).

• Since every group of order \(p^2\) is abelian, \(P \cong \ZZ_9\) or \(P \cong \ZZ_3 \times \ZZ_3\).

• Let \(Q = \langle y \rangle \cong \ZZ_2\) be a Sylow 2-subgroup. Since \(P \normal G\) and \(P \cap Q = \{1\}\), \(G \cong P \rtimes_\theta \ZZ_2\), where \(\theta: \ZZ_2 \to \Aut(P)\).

Case 1: \(P \cong \ZZ_9\). Here \(\Aut(\ZZ_9) \cong \ZZ_9^\times \cong \ZZ_6\), which has a unique element of order 2: the inversion automorphism \(x \mapsto x^{-1}\).

• If \(\theta\) is trivial: \(G \cong \ZZ_9 \times \ZZ_2 \cong \ZZ_{18}\).

• If \(\theta(y)\) is inversion (\(y x y^{-1} = x^{-1}\)): \(G \cong D_{18}\) (the dihedral group of order 18).

Case 2: \(P \cong \ZZ_3 \times \ZZ_3\). Here \(\Aut(\ZZ_3 \times \ZZ_3) \cong \GL_2(\FF_3)\). Up to conjugacy in \(\GL_2(\FF_3)\), there are two elements of order 2:

1. Trivial map: \(G \cong \ZZ_3 \times \ZZ_3 \times \ZZ_2 \cong \ZZ_3 \times \ZZ_6\).

2. Inversion on both coordinates (scalar \(-I = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}\)): \(y v y^{-1} = -v\) for all \(v \in P\). This gives \(G \cong (\ZZ_3 \times \ZZ_3) \rtimes \ZZ_2\), which is isomorphic to the generalized dihedral group \(D_9 \times \ZZ_3\) or \((\ZZ_3 \times \ZZ_3) \rtimes_{-1} \ZZ_2\).

3. Reflection (e.g. \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\)): \(y\) inverts one \(\ZZ_3\) factor and centralizes the other. This gives \(G \cong (\ZZ_3 \rtimes_{-1} \ZZ_2) \times \ZZ_3 \cong S_3 \times \ZZ_3\).

Thus, up to isomorphism, there are 5 groups of order 18:

1. \(\ZZ_{18}\) (abelian)

2. \(\ZZ_3 \times \ZZ_6\) (abelian)

3. \(D_{18}\)

4. \(S_3 \times \ZZ_3\)

5. \((\ZZ_3 \times \ZZ_3) \rtimes_{-1} \ZZ_2\)