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The group {1, −1} above and the cyclic group of order 3 under ordinary multiplication are both examples of abelian groups, and inspection of the symmetry of their Cayley tables verifies this. In contrast, the smallest non-abelian group, the dihedral group of order 6, does not have a symmetric Cayley table.
Integer multiplication respects the congruence classes, that is, a ≡ a' and b ≡ b' (mod n) implies ab ≡ a'b' (mod n). This implies that the multiplication is associative, commutative, and that the class of 1 is the unique multiplicative identity. Finally, given a, the multiplicative inverse of a modulo n is an integer x satisfying ax ≡ ...
In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system. The decimal multiplication table was traditionally taught as an essential part of elementary arithmetic around the world, as it lays the foundation for arithmetic operations ...
to implement multiplication as a sequence of table look ups for the log g (a) and g y functions and an integer addition operation. This exploits the property that every finite field contains generators. In the Rijndael field example, the polynomial x + 1 (or {03}) is one such generator.
It requires memorization of the multiplication table for single digits. This is the usual algorithm for multiplying larger numbers by hand in base 10. A person doing long multiplication on paper will write down all the products and then add them together; an abacus-user will sum the products as soon as each one is computed.
Tables Generator, a WYSIWYG table generator for Mediawiki markup; Wikitable Editor, a table editor of wiki code that outputs visual preview quickly; HTML2Wiki Converter at toolforge.org; HTML-WikiConverter, various versions and languages; pywikipediabot, can convert HTML tables to wiki; Table of CSS color names and HEX codes