Australian Mathematical Olympiads 1979-1995 by Hans Lausch, Peter Taylor

By Hans Lausch, Peter Taylor

This e-book is an entire number of all Australian Mathematical Olympiad papers from the 1st paper in 1979 to 1995. recommendations to the entire difficulties are incorporated and in a couple of situations, substitute ideas also are provided.

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8. C Proof. Straightforward computation. 9. [GQ(A,A,q),GQ(A,A)] = EQ(A,A,q). Proof. e. q =A) first. 5 shows that the left-hand side contains the right-hand side. Conversely, suppose Pl I = image of P in GQ 4n(A,A) ~ . 8implies (P1I)- 1 (11P)€EQ 4 n(A,A). Thus, if rr€GQ 2 n(A,A) we obtain (for suitable E , E 1 , E 2 € EQ 4 n(A, A)) (rr 1 I) (P 1 I) = (rr 1 I) (I 1 P) E = (11 P)(l1 rr) E 1 E = (11 (Prr))E 1 E = (Prr 1 I)E 2 E 1 E = (P 11) (rr 1 I) E 2 E 1 E. We consider the general case next. We use the relativization procedure described in §4C and identify G = GQ(Aixq,A1>< GQ(A,A,q) and E = EQ(Aixq,AtxAnq) with EQ(A,A)tx EQ(A,A,q).

The group [G, G) is the commutator subgroup of G. G is called connected or perfect if G = [G,G). 6. Details will be left to the reader. 7. If n ~ 3 then (a) EQ 2 n(A, A, q) is generated as a normal subgroup of EQ 2 n(A, A) by all where a and {3 (~ ~) and (~ ~) =0 mod q. (b) EQ 2 n(A, A, q) = [EQ 2 n(A, A, q), EQ 2 n(A, A)). In particular, EQ 2 n(A, A) is perfect. There is a natural embedding §3. AUTOMORPHISM GROUPS OF NONSINGULAR MODULES 31 We define - GQ(A,A,q) = lim GQ 2 n(A,A,q) n --- EQ(A,A,q) = lim EQ 2 n(A,A,q).

J) 0 d enote matnces · · to nonsingu· correspon d mg lar A-hermitian forms on respectively An and A 2 n. 12. 10d), there is an isomorphism Proof. The matrix (1 f3 ) {3-1 defines an isomorphism §3. AUTOMORPHISM GROUPS OF NONSINGULAR MODULES Let A(A, A) be a form ring. e. a= transpose (aij). If we pick a basis for the free right A-module An and the dual basis for (An) * then we can identify the group Aut (A-H(A n)) with a subgroup of GL 2 n(A) called the general A-quadratic group. 1. A 2nx2n matrix (a if and only ·if ~) f GL 2 n(A) belongs to GQ 2 n(A,A) y {3)-1 = o (-8 Xy ii) The diagonal coefficients of ya and 8(3 lie in A.

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