Fox derivative explained

In mathematics, the Fox derivative is an algebraic construction in the theory of free groups which bears many similarities to the conventional derivative of calculus. The Fox derivative and related concepts are often referred to as the Fox calculus, or (Fox's original term) the free differential calculus. The Fox derivative was developed in a series of five papers by mathematician Ralph Fox, published in Annals of Mathematics beginning in 1953.

Definition

\Z G which is denoted (\partial)/(\partial g _{i}) , and obeys the following axioms:(\partial)/(\partial g _{i}) (g _{j)} = \delta _{ij} , where \delta _{ij} is the Kronecker delta(\partial)/(\partial g _{i}) (e) = 0 (\partial)/(\partial g _{i}) (uv) = (\partial)/(\partial g _{i}) (u) + u (\partial)/(\partial g _{i}) (v) for any elements u and v of G.The first two axioms are identical to similar properties of the partial derivative of calculus, and the third is a modified version of the product rule. As a consequence of the axioms, we have the following formula for inverses(\partial)/(\partial g _{i}) (u ^{-1} ) = - u ^{-1} (\partial)/(\partial g _{i}) (u) for any element u of G.

Applications

The Fox derivative has applications in group cohomology, knot theory, and covering space theory, among other areas of mathematics.

See also

• Alexander polynomial
• Free group
• Ring (mathematics)
• Integral domain
• Derivation (differential algebra)

References

• Book: Brown, Kenneth S. . Kenneth Brown (mathematician) . Cohomology of Groups . . 1972 . 0-387-90688-6 . . 0672956 . 87 .
• Fox . Ralph . Ralph Fox . May 1953 . Free Differential Calculus, I: Derivation in the Free Group Ring . . 57 . 3 . 547–560 . 10.2307/1969736 . 1969736 . 0053938.
• Fox . Ralph . March 1954 . Free Differential Calculus, II: The Isomorphism Problem of Groups . Annals of Mathematics . 59 . 2 . 196–210 . 10.2307/1969686 . 1969686 . 0062125.
• Fox . Ralph . November 1956 . Free Differential Calculus, III: Subgroups . Annals of Mathematics . 64 . 2 . 407–419 . 10.2307/1969592 . 1969592 . 0095876.
• Chen . Kuo-Tsai . Kuo-Tsai Chen . Ralph Fox . Roger Lyndon . Roger Lyndon . July 1958 . Free Differential Calculus, IV: The Quotient Groups of the Lower Central Series . Annals of Mathematics . 68 . 1 . 81–95 . 10.2307/1970044 . 1970044 . 0102539.
• Fox . Ralph . May 1960 . Free Differential Calculus, V: The Alexander Matrices Re-Examined . Annals of Mathematics . 71 . 3 . 408–422 . 10.2307/1969936 . 1969936 . 0111781.