Introduction to finite group theory

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1. Nielsen’s method and generators of Aut.F n / 125 Nielsen automorphisms. An automorphism ˛ of F n is called monomial if there exists a permutation of 1;:::;n and i 2f 1; 1g such that ˛.x i / D x i i for i D 1;:::;n. 1.4 Exercise. Any monomial automorphism of the group F n is a composition of Nielsen automorphisms. Given an automorphism ˛ of F n and an element w 2 F n , we denote by w˛ the image of w under ˛. For any tuple of elements U D .u 1 ;:::;u m / of the group F n , we define U˛ D .u 1˛;:::;u m˛/. 1.5 Theorem. The group Aut.F n / is generated by the set of all Nielsen automorphisms n i and n ij . Proof. Let ˛ be an arbitrary automorphism of the group F n . By Theorem 1.2, the tuple U D X˛ D .u 1 ;:::;u n / can be carried by elementary Nielsen transformations into a Nielsen reduced tuple V . If one replaces the i-th entry of U by ui 1 or by u i u j for i ¤ j , then the new tuple U 0 has the form U 0 D Xn i˛ or U 0 D Xn ij ˛. By induction we get that V D Xˇ1 :::ˇs˛, where ˇ1;:::;ˇs are Nielsen automorphisms. Thus V is an automorphic image of X and hence V generates F n .In particular, any element x 2 X can be expressed in the form x D w 1 :::w k where w i 2 V [ V 1 and w i w iC1 ¤ 1. By Corollary 1.3 we obtain jxj > k. Therefore k D 1 and hence x 2 V [ V 1 . This implies that V D Xˇ0, where ˇ0 is a monomial automorphism. Then ˇ1 :::ˇs˛ D ˇ0, which, in view of Exercise 1.4, completes the proof of the theorem. 1.6 Definition. Let G be a group. For any element g 2 G, we define the inner automorphism i g of G by i g .x/ D gxg 1 for each x 2 G. The set fi g j g 2 Gg is a subgroup of Aut.G/. This is called the subgroup of inner automorphisms of G and it is denoted by Inn.G/. Clearly, Inn.G/ is isomorphic to the factor group of G by its center. For example, if G is a noncyclic free group, then Inn.G/ Š G. Clearly Inn.G/ Aut.G/. The group Out.G/ D Aut.G/=Inn.G/ is called the outer automorphism group of G. The image of an automorphism 2 Aut.G/ in the group Out.G/ is denoted by Œ and is called the outer automorphism corresponding to . 1.7 Theorem. Let F n be a free group with basis X D fx 1 ;x 2 ;:::;x n g. Let ‰ W Aut.F n / ! GL n .Z/ be the map defined by the following rule: for any ˛ 2 Aut.F n / the .i; j /-th entry of the matrix ‰.˛/ is equal to the sum of exponents of the letter x j in the word ˛.x i /. Then the map ‰ is an epimorphism. 31 Proof. Easy calculations show that ‰ is a homomorphism. This homomorphism is surjective since the images of the Nielsen automorphisms n ij and n i are transvections and diagonal matrices respectively, and they generate the group GL n .Z/. 31 The kernel of ‰ coincides with Inn.F n / for n D 2 and is strictly larger than Inn.F n / for n > 3 (see [40]).
126 Chapter 3. Automorphisms of free groups and train tracks Clearly the kernel of ‰ contains all inner automorphisms of F n . Therefore ‰ induces a homomorphism x‰ W Out.F n / ! GL n .Z/. For any O 2 Out.F n / the matrix x‰.O/ is called the abelianization matrix of O. 2 Maps of graphs. Tightening, collapsing and expanding 2.1 Graphs. Let G be a connected graph 32 . The sets of vertices and edges of G are denoted by V.G/ and E.G/. For any edge e 2 E.G/ the inverse edge is denoted by Ne. For any subset S E.G/ disjoint from the set S DfNe j e 2 Sg we write S ˙ D S [ S . A path in G is either a vertex of G (in this case the path is called trivial) or a nonempty sequence of edges e 1 e 2 :::e k , where the end of e i coincides with the beginning of e iC1 for 1 6 i 6 k 1. The initial vertex of a path p is denoted by ˛.p/, the terminal by !.p/, the inverse path to p is denoted by Np. A path is reduced if it does not contain a subpath of the form e Ne where e 2 E.G/. The definitions of homotopic paths and the fundamental group 1 .G; v/ are given in Section 4 of Chapter 2. The homotopy class of a path p is denoted by Œp. The set of all paths in G is denoted by P.G/. 2.2 Maps between graphs. A map from a graph G 1 to a graph G 2 is a map f W V.G 1 / [ E.G 1 / ! V.G 2 / [ P.G 2 / which sends V.G 1 / to V.G 2 /, E.G 1 / to P.G 2 / and preserves the relations of incidence and inverse. We write f W G 1 ! G 2 for ease. If v 1 is a distinguished vertex of G 1 and v 2 D f.v 1 /, we write f W .G 1 ;v 1 / ! .G 2 ;v 2 /. The map f can be extended naturally to paths in G 1 : f.e 1 e 2 :::e k / D f.e 1 /f .e 2 /:::f.e k /. Note that the f -image of a reduced path (even an edge) may be not reduced. 2.3 Induced homomorphisms and outer automorphisms. Let G 1 , G 2 be connected graphs and let f W .G 1 ;v 1 / ! .G 2 ;v 2 / be a map. Then f induces a homomorphism f W 1 .G 1 ;v 1 / ! 1 .G 2 ;v 2 / given by the rule Œp 7! Œf .p/. Now we consider the situation where G D G 1 D G 2 . Let f W .G; v/ ! .G; f .v// be a map. With each path x from v to f.v/we associate the homomorphism x W 1 .G; v/ ! 1 .G; v/ given by Œp 7! Œxf .p/x 1 . If x and y are two paths from v to f.v/, then clearly y D i Œyx 1 B x , where i Œq is the inner automorphism of 1 .G; v/ given by the rule: Œl 7! Œqlq 1 . Therefore, if x is an automorphism of 1 .G; v/, then y is one too. Moreover, the images of x and y in Out. 1 .G; x// coincide. We denote this common image by f ~ and say that the map f determines (or induces) the outer automorphism f ~ of 1 .G; v/. 2.4 Tightening, collapsing and expanding of maps. A map f W G ! G is said to be tight if for each edge e 2 E.G/ the path f.e/ is reduced. To any map 32 See Definition 1.4 in Chapter 2.
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Oleg Bogopolski Introduction to Gro
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Preface This English edition differ
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A tree in the neighborhood of Sproc
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2 Chapter 1. Introduction to finite
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176 Index Perron-Frobenius eigenval
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Introduction to finite group theory

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