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Multiplication Table


A multiplication table is an array showing the result of applying a binary operator to elements of a given set S. For example, the following table is the multiplication table for ordinary multiplication.

×12345678910
112345678910
22468101214161820
336912151821242730
4481216202428323640
55101520253035404550
66121824303642485460
77142128354249566370
88162432404856647280
99182736455463728190
10102030405060708090100

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 C_4^((1)), C_4^((2)), and C_4^((3))--by permuting the symbols used for the group elements (Cotton 1990, p. 11).

The first such table can be written as follows.

C_4^((1))1ABC
11ABC
AABC1
BBC1A
CC1AB

The multiplication table for a second representation C_4^((2)) may be obtained from C_4^((1)) by interchanging A and B.

C_4^((2))1ABC
11ABC
AA1CB
BBCA1
CCB1A

And finally, a multiplication table for the third representation C_4^((3)) can be obtained from C_4^((1)) by interchanging A and C.

C_4^((3))1ABC
11ABC
AAC1B
BB1CA
CCBA1

See also

Abstract Group, Binary Operator, Group, Truth Table

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References

Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, 1990.

Referenced on Wolfram|Alpha

Multiplication Table

Cite this as:

Weisstein, Eric W. "Multiplication Table." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MultiplicationTable.html

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