Altered books materials guide: Fabulous ideas to create by Laurie Goodson, Visit Amazon's Betsy McLoughlin Page, search

By Laurie Goodson, Visit Amazon's Betsy McLoughlin Page, search results, Learn about Author Central, Betsy McLoughlin, , Susan Pickering

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Xn }. Xm . The words [Xl, ... , Xm] with m > 2 are neither M- nor C-test ones. To see this, put Al = X3XIX3"1, A2 = x3x2x3"1, A 3 = [Xl, X2]. Then rank(AI, A 2 , A 3 ) = 3 and [[Al, A2], A3] = [X3[XI, X3]X3"1, [Xl, X2]] = [[X3' [Xl, X2]], [Xl, X2]]. Hence, upon setting 2, and, especially, C-test words in n letters for Fm with m, n > 1 is unclear.

1 ([356]). Let J be a right ideal of the free group ring ZF. 4) of the element u generate J as a right ideal of ZF. 3]). A. A. , Combinatorial Methods © Springer Science+Business Media New York 2004 36 3. Other Special Elements We focus here on ß-primitive elements of a free group F, where ß is the augmentation ideal of ZF. Another motivation for considering ß-primitive elements comes from commutative algebra. We recall some facts here very briefly. A vector (Pb ... ,Pm) of (Laurent) polynomials is called unimodular if PI, ...

Other Special Elements We are now going to show that these derivatives do not generate ß even modulo ß2. , that the n x n matrix of coefficients (its (i,j)th entry is the coefficient at (Xj - 1) in the decomposition of di(w) above) has determinant O. It is easy to see that this matrix (denote it by A = (aij)) is antisymmetrie, with zeros on the diagonal. The determinant of a matrix like that must be 0 if n is odd. Indeed, consider a summand a1,il a2,i2 ... an,i n in the decomposition of the determinant.

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