Multiplication Table

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.