This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Some (finite) simple groups MTH913SS11 J.I.Hall 28 January 2011 1 1 CFSG (1.1) Theorem. A finite simple group is isomorphic to one of: (1) a cyclic group of prime order p : C p ; (2) an alternating group: Alt( n ) ; (3) a classical group: PSL n ( q ) , PSp n ( q ) , PSU n ( q ) , P n ( q ) ; (4) an exceptional Lie type group 2 B 2 ( q ) , 3 D 4 ( q ) , E 6 ( q ) , 2 E 6 ( q ) , E 7 ( q ) , E 8 ( q ) , F 4 ( q ) , 2 F 4 ( q ) , G 2 ( q ) , 2 G 2 ( q ) ; (5) a sporadic simple group, of which there are twentysix. 22 2 Alternating groups The symmetric group on the set , denoted Sym() (or S in [ KS ]) is the group of all permutations (bijections) of . The corresponding alternating group , Alt() or A , is the subgroup of all even permutations of . We write Sym( n ) for Sym( { 1 , 2 ,...,n } ) and so forth. For finite, consider the realization of Sym() by rational permutation matrices: 7 P with P i, ( i ) = 1 , P i,j = 0 otherwise . The alternating group then consists of those permutation matrices with deter minant 1. (2.1) Theorem. Alt( n ) is simple for n 5 . 22 (2.2) Proposition. For all finite n 2 , the alternating group Alt( n ) is the unique subgroup of index 2 in Sym( n ) . Proof. Let A 1 = Alt( n ), and let A 2 be a subgroup of index 2 in Sym( n ). As they have index 2 in Sym( n ), both A 1 and A 2 are normal. Then A 1 A 2 /A 2 A 1 /A 1 A 2 . This group is of order at most 2 and is a quotient of the simple group A 1 . Therefore it has order 1 and A 1 = A 2 . 2 3 Classical groups The generic classical (Lie type) group is GL n ( R ), the group of units of the algebra M n ( R ) of n n matrices over the ring R with identity. (GL stands for general linear .) We restrict ourselves to the case R = F , a field. When F = F q is finite, we write instead GL n ( q ) (and continue this convention for other groups defined in terms of fields). The group GL n ( F ) has two obvious normal subgroups. (1) The determinant homomorphism maps GL n ( F ) onto F . Its kernel is the subgroup SL n ( F ) of determinant 1 matrices (the special linear group). (2) The nonzero scalar matrices form the center of GL n ( F ), a group Z again isomorphic to F . We set PGL n ( F ) = GL n ( F ) /Z , the projective general lin 2 ear group. The image of the special linear group SL n ( F ) in PGL n ( F ) is the projective special linear group PSL n ( F ) = SL n ( F ) Z/Z SL n ( F ) /Z SL n ( F ) . This group is generally simple. (3.1) Theorem. PSL n ( F ) is simple except for ( n,  F  ) = (2 , 2) , (2 , 3) . 22 In the special case F = F 2 , the multiplicative group F has order 1; so all the groups GL n (2), PGL n (2), SL n (2), and PSL n (2) are isomorphic....
View
Full
Document
 Spring '11
 NA

Click to edit the document details