Wikipedia:List of hoaxes on Wikipedia/Cayley-Newbirth operation matrix

In mathematics, a Cayley-Newbirth operation matrix outlines the effects of the composition of group operators on elements of a group.

Such matrices were first used by Arthur Cayley and James Newbirth to describe the effects of composing permutations in the symmetric group S3 of order 6. An easy example of how a Cayley-Newbirth operation matrix is constructed and used is with regard to the integers Z, an infinite cyclic group. Here the two defined binary operators are + and *, hence we have the following Cayley-Newbirth operation matrix:

+            *                     +     +             *                     *     *             *

This matrix indicates that the result of composing addition with any other operator except addition is that of a multiplication operation.

Cayley-Newbirth operation matrices are generally 2 &times; 2, though for an Abelian group, they may be but 1 &times; 2, since one of the operators is commutative, i.e. $$+* => *+$$, where $$=>$$ is the Bayleigh operator equivalence symbol.

It is easy to tell from a Cayley-Newbirth operation matrix whether a group is associative; if $$ab =// ba$$ for all operators $$a$$ and $$b$$ in a group, where $$=//$$ is the Bayleigh operator equivalence negation symbol, then the group in question is not associative. Otherwise, the group is associative.

For groups endowed with a well-defined ternary operator, the Cayley-Newbirth operation matrix must be extended to include a column for $$o*$$, the identity operator. If $$*o* => o**$$ for all operators $$*$$ in the group, then the group is ternary-complete, that is $$a * b * c$$ is defined and closed for all $$a, b, c$$ in the group.

Cayley-Newbirth operation matrices also appear in linear transformation theory in linear algebra, as well as in the theory of functional equations. It is possible to express the Hamiltonian of a time-independent mechanical system in terms of Cayley-Newbirth operation matrices.