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H. E. Rose A COURSE ON FINITE GROUPS Web Sections, Web Chapters and Solution Appendix Springer 322 TABLE OF Web Sections The material presented in this Web Site is additional to that given in the main text { A Course on Finite Groups. It consists of extra sections to some of the chapters with in most cases a set of supplementary problems, two extra chapters providing an introduction to group representation theory, and a long Appendix giving solutions, ranging from brief hints to complete answers, to all of the problems listed in the main text. Chapter, section, definition, theorem, problem and equation numbers all follow on from those given in the main text. In fact these Web Sections could be printed, and then slotted into the main text at the appropriate places allowing for the non-consecutive page numbers. Again it is a pleasure to thank Ben Fairbairn for commenting on and improving the text. Web Section 3:6 Representations of A5 .............................325 3.7 Problems 3W . .327 Web Section 4:6 The Transfer . 331 4.7 Group Presentation, Part 2 . 341 4.8 Problems 4W . ??? Web Section 5:4 Transitive and Primitive Permutation Groups, Iwasawa's Lemma ................................................................351 5.5 Problems 5W . .358 Web Section 6:5 Further Applications. Burnside's Normal Complement Theorem, Groups with Cyclic Sylow Subgroups . 361 6.6 Problems 5W . .367 Web Section 7:5 Infinite Abelian Groups. A Brief Introduction . .371 Web Section 9:4 Schur-Zassenhaus Theorem . 381 9.5 Problems 9W . ??? 324 Web Section 12.6 Simple Groups of Order less than 1000000, a second proof of the Simplicity of the Groups Ln(q), and a method for generating Steiner Systems for some Mathieu Groups ................... ..........389 12.7 Problem12W ....................................... .......396 Web Chapter 13 Representation and Character Theory ........401 13.1 Representations and Modules ......................... ....402 13.2 Theorems of Schur and Maschke ....................... ...408 13.3 Characters and Orthogonality Relations . ..412 13.4 Lifts and Normal Subgroups .......................... ....422 13.5 Problems13 ....................................... .......427 Web Chapter 14 Character Tables and Theorems of Burnside and Frobenius ................................................... .........433 14.1 Character Tables .................................. ....... 434 14.2 Burnside’s prqs-theorem ..................................440 14.3 Frobenius Groups .................................. .......444 14.4 Problems14 ....................................... .......455 14.5 Appendix on Algebraic Integers ........................ ...459 Web Solution Appendix Answers and Solutions, Problems 2 ........461 Problems 3 ........................................... .............472 Problems 4 ........................................... .............481 Problems 5 ........................................... .............489 Problems 6 ........................................... .............497 Problems 7 ........................................... .............508 Problems 8 ........................................... .............515 Problems 9 ........................................... .............521 Problems 10 .......................................... .............524 Problems 11 .......................................... .............532 Problems 12 .......................................... .............538 Problems A ........................................... ............544 Problems B ........................................... ............545 Subgroup lattice diagrams for Groups of Order 8, 12 or 16 ....... ..547 3.6 Representations of A5 325 3.6 Representations of A5 At the beginning of this chapter we noted that most groups have several distinct representations; to illustrate this fact we discuss here some repre- sentations of A5. In Section 3.2 we introduced A5 as the group of all even permutations on a five element set. It can also be (indirectly) specified as the only non-Abelian simple group up to isomorphism of order less than 100,1 see Problem 6.15. The group A5 has a number of presentations, three are as follows: 3 5 2 P1 : ha; b j a = b = (ab) = ei, 5 5 2 4 3 P2 : ha; b j a = b = (ab) = (a b) = ei, 3 3 3 2 2 2 P3 : ha; b; c j a = b = c = (ab) = (bc) = (ca) = ei. To show that each of these presentations do in fact define A5 we can argue as follows. Consider P1. We associate permutations in A5 (treating A5 as a per- mutation group) with each generator and then show that the corresponding relations hold. This shows that a copy of A5 is a factor of P1. Secondly, we show that P1 has 60 elements and this will be done in Problem 3.26. Similar arguments are required for P2 and P3. For P1 set a 7! (1; 4; 2) and b 7! (1; 2; 3; 4; 5); then ab = (1; 5)(3; 4): It is immediately clear that the relations of P1 are satisfied. For P2 we set a 7! (2; 1; 3; 4; 5) and b 7! (1; 2; 3; 4; 5); then ab = (1; 4)(3; 5) and a4b = (1; 3; 2); and for P3 we set a 7! (1; 2; 3); b 7! (1; 2; 4) and c 7! (1; 2; 5); note that (1; 2; j)(1; 2; k) = (1; k)(2; j) if j; k > 2 and j 6= k. The group A5 also has a number of matrix representations, we give two here and one more in Problem 3.27, further representation examples can be found in the Atlas (1985). First we show that SL2(4) is one such represen- tation. This group is defined as the set of all 2 × 2 matrices A with det A = 1 defined over the 4-element field F4, see Section 3.3. Let the elements of F4 be 0; 1; c and c + 1; where 1 + c + c2 = 0; and we work `modulo 2', that is 1 + 1 = c + c = 0 and c2 = c + 1; see Section 12.2. Using the presentation P3, we set 1 1 1 c + 1 1 c a 7! ; b 7! and c 7! ; 1 0 c 0 c + 1 0 1 In fact, it is the only non-Abelian simple group of order less than 168, see Chapter 12. 326 3 Group Constructions and Representations then c + 1 c + 1 c + 1 c c + 1 1 ab = ; bc = and ca = : 1 c + 1 c c + 1 c + 1 c + 1 We leave it as an exercise for the reader to show that these matrices satisfy the relations in P3, and that this system contains 60 matrices. Our second matrix representation is L2(5) which is defined as follows. Working over the 5-element field (that is working `modulo 5'), we take SL2(5) and factor out its centre. The process of forming a factor group is defined 1 0 −1 0 Chapter 4. The centre of SL2(5) is f 0 1 , 0 −1 g, see Problem 3.19; hence a b −a −b we work in SL2(5) and treat c d and −c −d as the `same' matrix. Formally, the elements of L2(5) are the cosets of Z(SL2(5)) in the group SL2(5). Note that −1 ≡ 4(mod 5) et cetera. Using the presentation P2 above, we set 1 0 4 1 a 7! and b 7! ; 3 1 1 3 then 4 1 4 1 ab 7! and a4b 7! ; 3 1 4 0 and it is a simple matter to show that the relations of P2 are satisfied. The choice of matrices and permutations above do satisfy the required conditions but are definitely not unique; as an exercise the reader should find some other examples. Another representation of A5 is as the rotational symmetry group of a dodecahedron. A dodecahedron is a regular solid structure with twelve regular equal-sized plane pentagonal faces with edges of equal length. This structure has three types of rotational symmetry: (i) rotation about a line drawn through the centres of opposite faces, this has order 5 as the faces are regular and have five edges; (ii) rotation about opposite vertices, in this structure three pentagonal faces meet at a vertex, and so this symmetry has order 3; and (iii) rotation about the centres of opposite edges, this has order 2. Now using the presentation P1 above, if we associate a symmetry of type (ii) with a and a symmetry of type (i) with b, then a symmetry of type (iii) is associated with ab. To see this the reader should obtain a model of an dodecahedron and try it! It is now easily seen that the relations of P1 are satisfied, and (with some patience) that there are 60 symmetries in all. An excellent film, made by the Open University for their second year course in pure mathematics, illustrates this isomorphism clearly. The group A5 can also be treated as the rotational symmetry group of a icosahedron. An icosahedron I is a regular solid with 20 identical equilateral triangular faces and it can be inscribed in a dodecahedron D by taking the vertices of I as the centres of the faces of D. This process is self-inverse for we can also inscribe a dodecahedron inside an icosahedron using the same method. This latter representation of A5 is the one that has been used by 3.7 Problems 3W 327 some chemists to describe the structure of the carbon molecule Carbon60, see page 24. The Atlas (1985) gives some more representations of A5, for example as unitary groups defined over the fields F16 or F25. Also this group occurs as maximal subgroups of a number of simple groups, for example A6;L2(k) where k = 11; 16; 19; 29 and 31, and the Janko group J2 (sometimes called the Hall-Janko group). Further details are given in Chapter 12, the Atlas (1985), and the `online' Atlas. 3.7 Problems 3W Problem 3.25 (An Example of a Free Group) A linear fractional trans- formation f of the complex plane C to itself is a map of the form az + b f (z) = where ad − bc 6= 0 for a; b; c; d 2 : a;b;c;d cz + d C (i) Show that the mapping a b 7! f c d a;b;c;d is a homomorphism (see Section 4.1) from GL2(C) to the group of all linear fractional transformations. 1 0 1 2 (ii) Show that the matrices A = 2 1 and B = 0 1 generate a free subgroup in the group of all linear fractional transformations.
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