By Roger C. Lyndon

From the reviews:

"This ebook (...) defines the limits of the topic now known as combinatorial team concept. (...)it is a substantial success to have centred a survey of the topic into 339 pages. This incorporates a giant and worthy bibliography; (over 1100 ÄitemsÜ). ...the publication is a precious and great addition to the literature, containing many effects no longer formerly on hand in a booklet. it's going to definitely develop into a customary reference." *Mathematical experiences, AMS, 1979*

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**Example text**

If I/! is a second endomorphism, there are two natural ways of applying I/! to V; the first yields VI/! •• ), while the second yields V = (VI' V Z , • •• ) where each Vi is obtained from the Uj in the same way as XiI/! was obtained from the Xj' that is, if XiI/! i(U I • UZ, • •• ). Inspection shows that VI/! is in fact the matrix for the composite ¢ I/! while V is that for I/! 0 ¢. If we identify endomorphisms with their matrices (of course keeping X fixed) the above discussion comes to nothing more than a rule for multiplying matrices.

We shall show that, for at least one choice of x, the transformation (P(x), x) decreases Iwi. X± I) < 0 whence some D(P(x), x) < O. In the remaining case some Pij contains none of 01 = a, a2 = a-I, b l = b, and b 2 = b- I . Since aiEAi = Pil + P i2 , we have ai E P ik for k =F j; similarly, b j E Phj for h =F i. Either i = k and h = j or else i =F k and h =F j; in either case, by (tt) we have and again some D(P(x), x) < O. Interchanging q and 1" if necessary we can suppose that x is a or a-I, and, by (**), that x = a.

It now follows that in the reduced form for w all the letters Yi remain, whence w i= 1. D t We postpone for the moment the uses to which Burns puts this proposition, and derive from it two of the results that Schreier first obtained by this method. The first is the Nielsen-Schreier subgroup theorem. 8. Every subgroup of a free group is free. D Let F be free with basis X and let H be a subgroup of F. By a partial Schreier Iransversal for H in F we mean a set T such that the cosets Ht for lET are distinct, and such that every initial segment of an element of T belongs to T.