sporadic-SS11

sporadic-SS11 - Some (finite) simple groups MTH913-SS11...

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Unformatted text preview: Some (finite) simple groups MTH913-SS11 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 twenty-six. 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....
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sporadic-SS11 - Some (finite) simple groups MTH913-SS11...

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