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Published 2005 Revised 2019
x | 2 | 4 | 6 | 8 |
2 | 4 | 8 | 2 | 6 |
4 | 8 | 6 | 4 | 2 |
6 | 2 | 4 | 6 | 8 |
8 | 6 | 2 | 8 | 4 |
x | 1 | 2 | 3 | 4 |
1 | 1 | 2 | 3 | 4 |
2 | 2 | 4 | 1 | 3 |
3 | 3 | 1 | 4 | 2 |
4 | 4 | 3 | 2 | 1 |
x | 1 | 3 | 5 | 7 |
1 | 1 | 3 | 5 | 7 |
3 | 3 | 1 | 7 | 5 |
5 | 5 | 7 | 1 | 3 |
7 | 7 | 5 | 3 | 1 |
I | Q | Q ² | Q ³ | |
I | I | Q | Q ² | Q ³ |
Q | Q | Q ² | Q ³ | I |
Q ² | Q ² | Q ³ | I | Q |
Q ³ | Q ³ | I | Q | Q ² |
I | V | H | R | |
I | I | V | H | R |
V | V | I | R | H |
H | H | R | I | V |
R | R | H | V | I |
Same | BTF | I-O | Both | |
Same | Same | BTF | I-O | Both |
BTF | BTF | Same | Both | I-O |
I-O | I-O | Both | Same | BTF |
Both | Both | I-O | BTF | Same |
Learn about the rules for a group and the different groups of 4 elements by doing some simple puzzles.
This article only skims the surface of Galois theory and should probably be accessible to a 17 or 18 year old school student with a strong interest in mathematics.
Explore the properties of some groups such as: The set of all real numbers excluding -1 together with the operation x*y = xy + x + y. Find the identity and the inverse of the element x.