# Huppert B.'s Endliche Gruppen PDF

By Huppert B.

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That is, here exists a surjective morphiim q5 : L -+ V of loops such that ker(q5)E Z2. In the remainder of this section assume L is a symplectic 2-loop with defining morphism q5 : L -+ V and ( 1 , ~ = ) ker(q5). Let n = dim(V). Observe Id Symplectic 2-loops (2) There exist 8 E 8 and an isomorphism a! : L -+ L(8) such that = (1,O). Ira! 1. 5. 4 we have a bijection 8 I-+ L(8) between the F-space 8 ( n )and the set of ail symplectic 2-loops defined on F x V. 5, we may take L = L(8) for some cocycle 0 and n = (1,O).

Observe that Aq = BqDq with Dq = ND(k), where q E k E A. Similarly Aq = BqEq with Eq = NE(k). Now there exists an automorphism a of A centralizing B and A / B with Da = E (cf. 3 in [FGT]). As a centralizes B , (Bq)a= Bq. Thus (Aq)a= (Bq)a(Dq)a= Aq. Define a on Y by (qa)a = q(aa) for a E A. As Aqa = Aq, this action is well defined. Then la = (qD)a = q(Da) = qE = m. As a commutes with B and A is the set of orbits of B on Y, a permutes A. So a E Aut(Y).

Therefore (3) holds and N+ = C N ( Z ) . Finally from the commutators in the previous paragraph, is centralized by +l(a)and [kl,&(a)] = k2, completing the proof of (2). E L}, and A = A 2 u A 3 . 4: (1) Q1 Nl = C N ( z l ) and kl € Z(N1). (2) Q1 Dg is edmspecial and c R I ( Q 1 )= (zl). (3) E2"+1 2 E+ 9 N . (4) Each g E Nl can be written uniquely in the fonn f o r d , e ~ L a, E r ; a n d g ~ Q 1i f and only i l e € ( n ) a n d a E ~. (5) K n Q1 = ( k l ) . (6) Q2nQ1= E f and N1/Q1 is the split extension of (Q2nNl)/E+Z v by $1 (r)/$l(El 2 ro.