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By Bergman G.M.

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An ) is an element of X, and a ∈A ∪ B is a nonidentity element, and does not come from the same group A or B from which a1 comes (which we take to be satisfied trivially if n =0), we define a(a1 , ... , an ) to be (a, a1 , ... , an ), which we see is indeed an element of X. There are three other possible relations between elements (a1 , ... , an ) ∈X and a ∈A ∪ B: (ii) If a = 1, we of course define a(a1 , ... , an ) = (a1 , ... , an ). (iii) If a ≠ 1 and this belongs to the same group as a1 , we define a(a1 , ...

6 [~]: Note that merely because the group F(S) is generated by the elements of S, every element of this group can be written in the form xiν1 ... xiν r as described, for some x1 , ... , xn ∈S 1 r and some family of exponents. This corollary tells us that every element of F(S) has a unique expression in this form. This is the explicit description of the free group promised earlier; it allows us to compute easily in such groups. 8 [~]: Here Lang finally gets to the point of the construction G1 ° ...

Lang’s terminology and notation differ in some small ways from what I have described: He writes ‘‘determined by generators and relations’’ in place of the standard ‘‘presented by generators and relations’’. 68) only considering relations of the form r = e, and calls the elements r the ‘‘relations’’. ) Then, on the next page, he passes to relations written as equations, as discussed above. 70. 3 on that page. If you do read the examples mentioned, note that the ‘‘exercise’’ referred to on the line following the first display is not trivial; cf.

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A companion to S.Lang's Algebra 4ed. by Bergman G.M.


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