Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.5 Exercise 1.5.2
Write out the group tables for $S_3$, $D_8$, and $Q_8$.
Solution:
Group table for $S_3$:
| 1 | (1 2) | (1 3) | (2 3) | (1 2 3) | (1 3 2) | |
| 1 | 1 | (1 2) | (1 3) | (2 3) | (1 2 3) | (1 3 2) |
| (1 2) | (1 2) | 1 | (1 2 3) | (1 3 2) | (1 3) | (2 3) |
| (1 3) | (1 3) | (1 3 2) | 1 | (1 2 3) | (2 3) | (1 2) |
| (2 3) | (2 3) | (1 2 3) | (1 3 2) | 1 | (1 2) | (1 3) |
| (1 2 3) | (1 2 3) | (2 3) | (1 2) | (1 3) | (1 3 2) | 1 |
| (1 3 2) | (1 3 2) | (1 3) | (2 3) | (1 2) | 1 | (1 2 3) |
Group table for $D_8$:
| $1$ | $r$ | $r^2$ | $r^3$ | $s$ | $sr$ | $sr^2$ | $sr^3$ | |
| $1$ | $1$ | $r$ | $r^2$ | $r^3$ | $s$ | $sr$ | $sr^2$ | $sr^3$ |
| $r$ | $r$ | $r^2$ | $r^3$ | $1$ | $sr^3$ | $s$ | $sr$ | $sr^2$ |
| $r^2$ | $r^2$ | $r^3$ | $1$ | $r$ | $sr^2$ | $sr^3$ | $s$ | $sr$ |
| $r^3$ | $r^3$ | $1$ | $r$ | $r^2$ | $sr$ | $sr^2$ | $sr^3$ | $s$ |
| $s$ | $s$ | $sr$ | $sr^2$ | $sr^3$ | $1$ | $r$ | $r^2$ | $r^3$ |
| $sr$ | $sr$ | $sr^2$ | $sr^3$ | $s$ | $r^3$ | $1$ | $r$ | $r^2$ |
| $sr^2$ | $sr^2$ | $sr^3$ | $s$ | $sr$ | $r^2$ | $r^3$ | $1$ | $r$ |
| $sr^3$ | $sr^3$ | $s$ | $sr$ | $sr^2$ | $r$ | $r^2$ | $r^3$ | $1$ |
Group table for $Q_8$:
| $1$ | $-1$ | $i$ | $-i$ | $j$ | $-j$ | $k$ | $-k$ | |
| $1$ | $1$ | $-1$ | $i$ | $-i$ | $j$ | $-j$ | $k$ | $-k$ |
| $-1$ | $-1$ | $1$ | $-i$ | $i$ | $-j$ | $j$ | $-k$ | $k$ |
| $i$ | $i$ | $-i$ | $-1$ | $1$ | $k$ | $-k$ | $-j$ | $j$ |
| $-i$ | $-i$ | $i$ | $1$ | $-1$ | $-k$ | $k$ | $j$ | $-j$ |
| $j$ | $j$ | $-j$ | $-k$ | $k$ | $-1$ | $1$ | $i$ | $-i$ |
| $-j$ | $-j$ | $j$ | $k$ | $-k$ | $1$ | $-1$ | $-i$ | $i$ |
| $k$ | $k$ | $-k$ | $j$ | $-j$ | $-i$ | $i$ | $-1$ | $1$ |
| $-k$ | $-k$ | $k$ | $-j$ | $j$ | $i$ | $-i$ | $1$ | $-1$ |
