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Date: 12-2-2020
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A multiplication table is an array showing the result of applying a binary operator to elements of a given set . For example, the following table is the multiplication table for ordinary multiplication.
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
2 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |
3 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 |
4 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 |
5 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 |
6 | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 |
7 | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 |
8 | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 |
9 | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 |
10 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |
The results of any binary mathematical operation can be written as a multiplication table. For example, groups have multiplication tables, where the group operation is understood as multiplication. However, different labelings and orderings of a multiplication table may describe the same abstract group. For example, the multiplication table for the cyclic group C4 may be written in three equivalent ways--denoted here by , , and --by permuting the symbols used for the group elements (Cotton 1990, p. 11).
The first such table can be written as follows.
1 | ||||
1 | 1 | |||
1 | ||||
1 | ||||
1 |
The multiplication table for a second representation may be obtained from by interchanging and .
1 | ||||
1 | 1 | |||
1 | ||||
1 | ||||
1 |
And finally, a multiplication table for the third representation can be obtained from by interchanging and .
1 | ||||
1 | 1 | |||
1 | ||||
1 | ||||
1 |
REFERENCES:
Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, 1990.
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