Fox derivative

If G is a free group with identity element e and generators g_i, then the Fox derivative with respect to g_i is a function from G into the integral group ring ${\displaystyle \mathbb {Z} G}$ which is denoted ${\displaystyle {\frac {\partial }{\partial g_{i}}}}$, and obeys the following axioms:

• ${\displaystyle {\frac {\partial }{\partial g_{i}}}(g_{j})=\delta _{ij}}$, where ${\displaystyle \delta _{ij}}$ is the Kronecker delta
• ${\displaystyle {\frac {\partial }{\partial g_{i}}}(e)=0}$
• ${\displaystyle {\frac {\partial }{\partial g_{i}}}(uv)={\frac {\partial }{\partial g_{i}}}(u)+u{\frac {\partial }{\partial g_{i}}}(v)}$ 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

• ${\displaystyle {\frac {\partial }{\partial g_{i}}}(u^{-1})=-u^{-1}{\frac {\partial }{\partial g_{i}}}(u)}$ for any element u of G.

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

• Alexander polynomial
• Free group
• Ring (mathematics)
• Integral domain

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