By Vladimir Shpilrain
the most objective of this booklet is to teach how principles from combinatorial workforce thought have unfold to 2 different parts of arithmetic: the idea of Lie algebras and affine algebraic geometry. a few of these rules, in flip, got here to combinatorial workforce idea from low-dimensional topology at the start of the twentieth Century.
This ebook is split into 3 quite self reliant components. half I presents a quick exposition of a number of classical ideas in combinatorial workforce conception, particularly, equipment of Nielsen, Whitehead, and Tietze. half II includes the focus of the publication. right here the authors convey how the aforementioned ideas of combinatorial crew thought discovered their means into affine algebraic geometry, a desirable quarter of arithmetic that stories polynomials and polynomial mappings. half III illustrates how rules from combinatorial workforce conception contributed to the idea of loose algebras. the point of interest here's on Schreier kinds of algebras (a number of algebras is related to be Schreier if any subalgebra of a unfastened algebra of this sort is loose within the comparable number of algebras).
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Extra resources for Combinatorial Methods: Free Groups, Polynomials, and Free Algebras
Suppose M is invertible over ZFj denote by U the row matrix (Ul. ,um ) and by Y the row matrix (yl. 'Ym). Then UMM- 1 = Y M- 1 = U, which means that Ul. ,Um belong to the right ideal of ZF generated by Yl. ,Ym' Conversely, suppose we have Y B = U for some matrix B over ZF. Then UMB = U, and hence MB = I, the identity matrix, because (Ul. , um) form a free basis of a free right ZF-module J. , . 2, we consider a couple of examples. 10. The element u = [Xl. X2][X3, X4] . [X2m-1. 4. 11. The element v = [Xl.
3. 3. A nontrivial word w(XI, ... , x n ) is aC-test word in n letters for Fm if for any two n-tuples (Al, ... , An), (Bl, ... , B n ) of elements of Fm the equality w(AI, ... , An) = w(B I , ... , B n ) =f. 1 implies the existence of an element S E Fm such that Bi = SAiS- 1 for all i = 1,2, ... 3. First, any C-test word in n letters is both a test and M-test element for Fn . 3 is the most restrictive one. The commutator [Xl, X2] is an M-test word (for F2 ) but not aC-test word in two letters. If aC-test word w in n letters is not a proper power, then the stabilizer of w in AutFn is (Tw ).
D. Lee , based on ideas of S. Ivanov , confirmed the following conjecture of Shpilrain [42, Problem (F7)]: every endomorphism of a free group of finite rank is completely determined by its values on just two (explicitly given) elements unless the image of this endomorphism is cydic. We therefore have a test set of two elements that allows one to distinguish any two endomorphisms of a free group that have noncydic images. 1 Nielsen's Commutator Test Nielsen  gave the following commutator test for an endomorphism of the free group F = F 2 = (x, y) to be an automorphism: an endomorphism
Combinatorial Methods: Free Groups, Polynomials, and Free Algebras by Vladimir Shpilrain