**Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.5 Exercise 1.5.1**

Compute the order of each of the elements in $Q_8$.

Solution:

$x$ | Reasoning | Order |
---|---|---|

1 | 1 is the identity. | 1 |

-1 | $(-1) \cdot (-1) = 1$ | 2 |

$i$ | $i \cdot i=-1$, $(-1) \cdot i = -i$, $(-i) \cdot i = 1$ | 4 |

$-i$ | $(-i) \cdot (-i)=-1$, $(-1) \cdot (-i) = i$, $i \cdot (-i) = 1$ | 4 |

$j$ | $j \cdot j = -1$, $(-1) \cdot j = -j$, $(-j) \cdot j = 1$ | 4 |

$-j$ | $(-j) \cdot (-j) = -1$, $(-1) \cdot (-j) = j$, $i \cdot (-j) = 1$ | 4 |

$k$ | $k \cdot k = -1$, $(-1) \cdot k = -k$, $(-k) \cdot k = 1$ | 4 |

$-k$ | $(-k) \cdot (-k) = -1$, $(-1) \cdot (-k) = k$, $k \cdot (-k) = 1$ | 4 |