Introduction to finite group theory

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15. A theorem of Jordan and Dickson 23 following: 0 1 ; 1 0 ; 1 1 ; 1 x ; 1 y ; and no two of these vectors are scalar multiples of each other. Therefore V contains exactly five one-dimensional subspaces (lines). As above the group PSL 2 .4/ acts faithfully on the set of these lines. Consider the elements xA and xB of the group PSL 2 .4/ which are the images of the matrices A D xy x0 and B D 11 10 . It is easy to verify that for an appropriate numeration of lines the element xA acts on them as the permutation (12345), and the element xB as the permutation (123). In Section 12 we showed that h.12345/; .123/i DA 5 . This enables us to complete the proof as in part 1). 14.2 Exercise. PSL 2 .2/ Š S 3 , PSL 2 .3/ Š A 4 . 15 A theorem of Jordan and Dickson 15.1 A simplicity criterion. Let a group G act faithfully and 2-transitively on the set X. Suppose that the following conditions hold: 1) G coincides with its commutator G 0 ; 2) in the stabilizer St.x/ of an element x 2 X there is a subgroup A such that a) A is abelian, b) A St.x/, c) G DhgAg 1 j g 2 Gi. Then G is simple. Proof. Let N be a nontrivial normal subgroup of the group G. By Proposition 6.7, N acts transitively on X and hence G D N St.x/. We will prove that G D NA. By condition c), every element g from G can be written as g D g 1 a 1 g1 1 :::g ka k g 1 k , where a i 2 A, g i 2 G. Write each g i as g i D n i s i , where n i 2 N , s i 2 St.x/. Then the image of the element g in G=N coincides with the image of the element a D s 1 a 1 s1 1 :::s ka k s 1 . By condition b) we have a 2 A and so g 2 Na NA. k Finally, G D G 0 D .NA/ 0 6 N since modulo N the commutator Œn 1 a 1 ;n 2 a 2 , where n i 2 N , a i 2 A, is equal to the commutator Œa 1 ;a 2 , i.e., equal to 1 because of the commutativity of A. 15.2 Theorem (Jordan–Dickson). Let K be a field, n > 2. The group PSL n .K/ is simple with the exception of PSL 2 .2/ and PSL 2 .3/. Proof. Let V be the vector space consisting of all columns of size n over the field K. Let e 1 ;:::;e n be the standard basis of V . Let X be the set of all one-dimensional subspaces of V (lines). For any nonzero vector v 2 V denote by Nv the line in X
24 Chapter 1. Introduction to finite group theory containing v. For any matrix M 2 SL n .K/ denote by SM the image of M in the group G D PSL n .K/. We define the action of the group G on the set X by the rule SM Nv D Mv and prove that this action satisfies the conditions of the simplicity criterion. First we will prove that G acts faithfully on X. Suppose not, then there exists SM 2 G such that SM stabilizes each line from X. Then Me i D i e i and M.e 1 CCe n / D .e 1 CCe n / for some i and in K. By linearity we deduce that 1 DD n D . Therefore M is a scalar matrix and SM D 1. Thus the action of G on X is faithful. This action is 2-transitive, since the lines Ne 1 and Ne 2 can be carried to any two different lines Nv 1 and Nv 2 by the element SM , where M is a matrix from SL n .K/ with the first and the second columns multiples of v 1 and v 2 respectively. Now we prove that .PSL n .K// 0 D PSL n .K/. In general, from N H and H D H 0 it follows that .H=N / 0 D H=N. Therefore it is sufficient to prove that .SL n .K// 0 D SL n .K/. For n > 3 this follows from the claims 1) and 2) of Exercise 15.3, and for n D 2 from the claims 1) and 3), and the fact that for jKj >3 the group SL 2 .K/ contains a non-scalar diagonal matrix. Let x be the line containing the vector e n . Its stabilizer St.x/ in the group G consists of all xB such that the column Be n is a multiple of the column e n , that is, St.x/ DfxB j B 1n DDB n 1;n D 0g. Let A be the subgroup of the group St.x/ consisting of all xB such that B differs from the identity matrix only by elements in the positions .n;1/;:::;.n;n 1/. It is easy to verify that A is an abelian group and A St.x/. The equation PSL n .K/ DhgAg 1 j g 2 PSL n .K/i follows from the fact that A contains the images of the transvections of the form t ni .˛/ and from the claims 1) and 4) of the following exercise. 15.3 Exercise. 1) The group SL n .K/ is generated by all its transvections t ij .˛/. 2) Œt ik .˛/; t kj .ˇ/ D t ij .˛ˇ/ for distinct i, j , k. d 3) Œt ij .˛/; d D t ij ˛ 1 i d j , where d is a diagonal matrix from GLn .K/ with elements d 1 ;:::;d n on the main diagonal. 4) M t ij .˛/M 1 D t .i/.j/ .˛/, where 2 S n and M is the matrix with 1 in the positions ..1/; 1/;:::;..n/; n/ and 0 in the other positions. How will this formula be changed if in M we replace one of the 1’s by 1 15.4 Remarks. 1) The groups PSL n .q/ are simple for n > 2 and .n; q/ ¤ .2; 2/; .2; 3/, they belong to the family of finite simple groups of Lie type (see [19], [25]). With a finite number of exceptions they are not isomorphic to any alternating group. In most cases this follows just by comparing their orders. For example, j PSL 2 .7/j D168 ¤jA m j for all m. We notice the following surprising isomorphisms: PSL 2 .4/ Š PSL 2 .5/ Š A 5 ; PSL 2 .7/ Š PSL 3 .2/;
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