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

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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 × 2, though for an Abelian group, they may be but 1 × 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.