Section 14 Groups of Small Order

Let \(|\mathsf{G}| = n\).

14.1 n = 1

The only group of order 1 is the trivial group \(G = \{ e \}\).

14.2 n = 2

Since 2 is prime the only group of order 2, up to isomorphism, is \(\mathbb{Z}_2\).

\[ \begin{array}{c|ccc} \mathbb{Z}_2 & 0 & 1 \\ \hline 0 & 0 & 1 \\ 1 & 1 & 0 \\ \end{array} \]

14.3 n = 3

Since 3 is prime the only group of order 3, up to isomorphism, is \(\mathbb{Z}_3\).

\[ \begin{array}{c|ccc} \mathbb{Z}_3 & 0 & 1 & 2 \\ \hline 0 & 0 & 1 & 2 \\ 1 & 1 & 2 & 0 \\ 2& 2 & 0 & 1 \\ \end{array} \]

14.4 n = 4

You proved that there are two groups of order 4. They are the cyclic group of order 4 and the Klein-4 group \[ \begin{array}{c|ccc} C_4\cong\mathbb{Z}_4 & 1 & a & a^2 & a^3 \\ \hline 1 & 1 & a & a^2 & a^3 \\ a & a & a^2 & a^3 & 1 \\ a^2 & a^2 & a^3 & 1 & a \\ a^3 & a^3 & 1 & a & a^2 \end{array} \qquad \begin{array}{c|ccc} K_4 & 1 & a & b & c \\ \hline 1 & 1 & a & b & c \\ a & a & 1 & c & b \\ b & b & c & 1 & a \\ c & c & b & a & 1 \end{array} \]

14.5 n = 5

Since 3 is prime the only group of order 5, up to isomorphism, is \(\mathbb{Z}_5\).

14.6 n = 6

We have the cyclic, and thus abelian group, \(\mathbb{Z}_6\). Furthermore, you proved in class that \(\mathbb{Z}_6 \cong \mathbb{Z}_3 \times \mathbb{Z}_2\).

We also have \(\mathsf{S}_3 \cong \mathsf{D}_3\).

14.7 n = 7

Since 3 is prime the only group of order 7, up to isomorphism, is \(\mathbb{Z}_7\).

14.8 n = 8

14.8.1 Abelian groups

We have the cyclic group of order 8: \(\mathbb{Z}_8\). And, we’ve seen, , by comparing orders of elements, that the following abelian groups of order 8, groups are not isomorphic to one another $_4 _2, _2 _2 _2. $

14.8.2 The Dihedral Group \(\mathsf{D}_4\)

14.8.3 The Quaternion Group \(Q_8\)

\[ \begin{array} {| r || c | c | c | c | c | c | c | c |} \hline Q_8 & \phantom{\Big\vert} \ 1 \ \phantom{\Big\vert} &\ -1 \ &\ \ {\mathbf{i}}\ \ &\ -{\mathbf{i}}\ &\ \ {\mathbf{k}}\ \ &\ -{\mathbf{k}}&\ \ {\mathbf{k}}\ \ &\ -{\mathbf{k}}\ \\ \hline\hline 1 \phantom{\Big\vert} & 1 & -1 & {\mathbf{i}}& - {\mathbf{i}}& {\mathbf{k}}& -{\mathbf{k}}& {\mathbf{k}}& - {\mathbf{k}}\\ \hline -1 \phantom{\Big\vert} & -1 & 1 & -{\mathbf{i}}& {\mathbf{i}}& -{\mathbf{k}}& {\mathbf{k}}& -{\mathbf{k}}& {\mathbf{k}}\\ \hline {\mathbf{i}}\phantom{\Big\vert} & {\mathbf{i}}& -{\mathbf{i}}& -1 & 1 & {\mathbf{k}}& -{\mathbf{k}}& -{\mathbf{k}}& {\mathbf{k}}\\ \hline -{\mathbf{i}}\phantom{\Big\vert} & -{\mathbf{i}}& {\mathbf{i}}& 1 & -1 & -{\mathbf{k}}& {\mathbf{k}}& {\mathbf{k}}& -{\mathbf{k}}\\ \hline {\mathbf{k}}\phantom{\Big\vert} & {\mathbf{k}}& -{\mathbf{k}}& -{\mathbf{k}}& {\mathbf{k}}& -1& 1& {\mathbf{i}}& - {\mathbf{i}}\\ \hline %&&&&&&&& \\ \hline -{\mathbf{k}}\phantom{\Big\vert} & -{\mathbf{k}}& {\mathbf{k}}& {\mathbf{k}}& -{\mathbf{k}}& 1& -1& -{\mathbf{i}}& {\mathbf{i}}\\ \hline {\mathbf{k}}\phantom{\Big\vert} & {\mathbf{k}}& -{\mathbf{k}}& {\mathbf{k}}& - {\mathbf{k}}& -{\mathbf{i}}& {\mathbf{i}}& -1 & 1 \\ \hline -{\mathbf{k}}\phantom{\Big\vert} & -{\mathbf{k}}& {\mathbf{k}}& -{\mathbf{k}}& {\mathbf{k}}& {\mathbf{i}}& -{\mathbf{i}}& 1 & -1 \\ \hline \end{array} \]

14.9 n = 12

One group of order 12 that we have seen is \({\mathsf{A}}_4\).

\[ {\tiny \begin{array}{c|cccccccccccc} {\mathsf{A}}_4& 1 & (1 2)(3 4) & (1 3)(2 4) & (1 4)(2 3) & (2 34) & (2 4 3) & (1 2 3) & (1 2 4) & (1 3 2) & (1 34) & (1 4 2) & (1 4 3) \\ \hline 1& 1 & (1 2)(3 4) & (1 3)(2 4) & (1 4)(2 3) & (2 34) & (2 4 3) & (1 2 3) & (1 2 4) & (1 3 2) & (1 34) & (1 4 2) & (1 4 3) \\ (1 2)(3 4) &(1 2)(3 4) & 1 & (14)(2 3) & (1 3)(2 4) & (1 2 4) & (1 2 3) & (2 43) & (2 3 4) & (1 4 3) & (1 4 2) & (1 3 4) & (1 32) \\ (1 3)(2 4) &(1 3)(2 4) & (1 4)(2 3) & 1 & (1 2)(34) & (1 3 2) & (1 3 4) & (1 4 2) & (1 4 3) & (2 34) & (2 4 3) & (1 2 3) & (1 2 4) \\ (1 4)(23) &(1 4)(23) & (1 3)(2 4) & (1 2)(3 4) & 1 & (1 4 3) & (1 42) & (1 3 4) & (1 3 2) & (1 2 4) & (1 2 3) & (2 43) & (2 3 4) \\ (2 3 4) &(2 3 4) & (1 3 2) & (1 4 3) & (1 24) & (2 4 3) & 1 & (1 3)(2 4) & (1 3 4) & (1 42) & (1 4)(2 3) & (1 2)(3 4) & (1 2 3) \\ (2 43) &(2 43) & (1 4 2) & (1 2 3) & (1 3 4) & 1 & (2 3 4) & (14 3) & (1 4)(2 3) & (1 2)(3 4) & (1 2 4) & (1 32) & (1 3)(2 4) \\ (1 2 3) &(1 2 3) & (1 3 4) & (2 43) & (1 4 2) & (1 2)(3 4) & (1 2 4) & (1 3 2) & (1 3)(2 4) & 1 & (2 3 4) & (1 4 3) & (1 4)(23) \\ (1 2 4) &(1 2 4) & (1 4 3) & (1 3 2) & (2 3 4) & (1 2 3) & (1 2)(3 4) & (1 4)(2 3) & (1 4 2) & (1 34) & (1 3)(2 4) & 1 & 2 4 3) \\ (1 3 2) &(1 3 2) & (2 34) & (1 2 4) & (1 4 3) & (1 3 4) & (1 3)(24) & 1 & 2 4 3) & (1 2 3) & (1 2)(3 4) & (1 4)(23) & (1 4 2) \\ (1 3 4) &(1 3 4) & (1 2 3) & (1 4 2) & (2 43) & (1 3)(2 4) & (1 3 2) & (1 2 4) & (1 2)(34) & (1 4)(2 3) & (1 4 3) & (2 3 4) & 1 \\ (1 42) &(1 42) & (2 4 3) & (1 3 4) & (1 2 3) & (1 4)(2 3) & (1 4 3) & (2 3 4) & 1 & (1 3)(2 4) & (1 3 2) & (1 24) & (1 2)(3 4) \\ (1 4 3) &(1 4 3) & (1 2 4) & (2 34) & (1 3 2) & (1 4 2) & (1 4)(2 3) & (1 2)(34) & (1 2 3) & (2 4 3) & 1 & (1 3)(2 4) & (1 3 4) \\ \end{array}} \]