Endomorphism monoid of Q8
This article is about a particular monoid, i.e., a field unique up to isomorphism. View a complete list of particular monoids
Definition
This monoid is the set of endomorphisms of the quaternion group, under composition.
Elements
This monoid has 28 elements. They are listed in the table below, with the heading denoting each element of the quaternion group, each cell in the other rows indicating where the element is mapped to under each of the 28 maps.
| 1 | -1 | i | -i | j | -j | k | -k |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 |
| 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 |
| 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 |
| 1 | -1 | i | -i | j | -j | k | -k |
| 1 | -1 | i | -i | -j | j | -k | k |
| 1 | -1 | i | -i | k | -k | -j | j |
| 1 | -1 | i | -i | -k | k | j | -j |
| 1 | -1 | -i | i | j | -j | -k | k |
| 1 | -1 | -i | i | -j | j | k | -k |
| 1 | -1 | -i | i | k | -k | j | -j |
| 1 | -1 | -i | i | -k | k | -j | j |
| 1 | -1 | j | -j | i | -i | -k | k |
| 1 | -1 | j | -j | -i | i | k | -k |
| 1 | -1 | j | -j | k | -k | i | -i |
| 1 | -1 | j | -j | -k | k | -i | i |
| 1 | -1 | -j | j | i | -i | k | -k |
| 1 | -1 | -j | j | -i | i | -k | k |
| 1 | -1 | -j | j | k | -k | -i | i |
| 1 | -1 | -j | j | -k | k | i | -i |
| 1 | -1 | k | -k | i | -i | j | -j |
| 1 | -1 | k | -k | -i | i | -j | j |
| 1 | -1 | k | -k | j | -j | -i | i |
| 1 | -1 | k | -k | -j | j | i | -i |
| 1 | -1 | -k | k | i | -i | -j | j |
| 1 | -1 | -k | k | -i | i | j | -j |
| 1 | -1 | -k | k | j | -j | i | -i |
| 1 | -1 | -k | k | -j | j | -i | i |