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By Roger C. Lyndon

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"This booklet (...) defines the limits of the topic now referred to as combinatorial crew thought. (...)it is a substantial fulfillment to have centred a survey of the topic into 339 pages. This contains a gigantic and necessary bibliography; (over 1100 ÄitemsÜ). ...the e-book is a invaluable and great addition to the literature, containing many effects now not formerly to be had in a ebook. it is going to unquestionably turn into a customary reference." Mathematical reports, AMS, 1979

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The natural map from Aut(F) into Aut(F) is an epimorphism. 0 A. W. Mostowski has proved the analogous result where F is replaced by the free group of an arbitrary variety; see P. M. Neumann (1965). 25 4. Automorphisms of Free Groups The kernel of this map clearly contains the group of inner automorphisms of F, but for n > 2, is larger, since, for example, the automorphism ex:X2 t-+ Xi tX 2X t, X3 t-+ X2! X3X2 is clearly not an inner automorphism. The following proposition is due to Nielsen (1917, see also Chang, 1960).

O The derived series of a free group has trivial intersection. ° If the free group F has rank or 1, then [F, F] = I and the assertion is trivial. If F has rank at least 2, then [F, F] is a free group of rank at least 2. It follows by induction that all terms of the derived series have rank at least 2. If F has rank at least 2, then F is nonabelian while FI[F, F] is abelian, whence [F, F] < F. 3) applies to give the conclusion. See also Kurosh (1955, 2nd. , §36). 10) below. H. Hopf (1932) asked whether a finitely generated group could be isomorphic to a proper quotient group of itself.

Now IF:HI finite implies that IF:GI and IG:HI are finite. 9) we have rank(G) - 1 IF:GI = rank(F) _ 1 ~ I, whence rank (G) > 1. 9) again we have IG: HI = rank(H) - 1 . rank(G) _ 1 ~ I, whence rank(G) ~ rank(H) wIth equahty only if IG:HI = I, that is, if G = H. 0 The next two propositions are from Karrass and Solitar (1969). 17. If H is a finitely generated subgroup of a free group F that has 20 Chapter I. Free Groups and Their Subgroups non-trivial intersection with every non-trivial normal subgroup of F, then H has finite index in F.

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