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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.


If G is a free group with identity element e and generators gi, then the Fox derivative with respect to gi is a function from G into the integral group ring ZG which is denoted \( \frac{\partial}{\partial g_i} \), and obeys the following axioms:

\( \frac{\partial}{\partial g_i}(g_j) = \delta_{ij} \), where \delta_{ij} \) is the Kronecker delta
\( \frac{\partial}{\partial g_i}(e) = 0 \)
\( \frac{\partial}{\partial g_i}(uv) = \frac{\partial}{\partial g_i}(u) + u\frac{\partial}{\partial g_i}(v) \) 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

\( \frac{\partial}{\partial g_i}(u^{-1}) = -u^{-1}\frac{\partial}{\partial g_i}(u) \) 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.
See also

Alexander polynomial
Free group
Ring (mathematics)
Integral domain


Brown, Kenneth S. (1972). Cohomology of Groups. Graduate Texts in Mathematics 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.

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