30 M . L. RACINE
(
o Q
Q 0
which is a maximal order of K0 by Lemma 3. By Lemma 2, P is a maximal
2 ' n
order of K_ . Hence, by Proposition 2, M = % n © is maximal. Denote J D
2n * n
by © ; by Lemma 3 any maximal order of K is isomorphic to P for some
o -ideal Q .
PROPOSITION 4. Let C = K the split quaternions, # = M(C ). Then
any maximal order M of J can be written M = J n E(L) where
n
L = 0 ) (ox. 0 ay.), a an o-ideal and {x.,y. } a symplectic bas e of v
i=l
a 2n dimensional K vector spac e (on which we let C act). M is
/ 0 Q
isomorphic to J n O where © =
° n
, a maximal order of C. Denote
V o o
this M by M ; M = M if and only if 0 = © (which by Lemma 5 is
Q Q Q Q Q
1 2 1 2
if and only if the clas s of a = the clas s of a in Q/0 ). Therefore the
isomorphism c l a s s e s of maximal orders of ? are in one-to-on e correspondence
2
with elements of the group Q/Q , where Q is the clas s group of K.
PROOF. Only the result s on isomorphisms remain to be shown.
Clearly if two maximal orders of C, P and .0 are isomorphic then
n n
£ n © = ^ n © Let L = 0 7 (ox. 0 a y . ) , L = 0 J (ov. 0 a w . ) ,
II 11 J. —' 1 A ' 1 £ r
1
1
Cd
1
(
x
. , y . } , {v.,w. } symplectic b a s e s of V, M, = ^ n E(L.) i = 1,2. An
isomorphism of M onto M induces an automorphism of #= KM = KM
which by Martindale's Theorem extends to the algebra with involution
(C , J ) = (K , J ). Such automorphisms are of the form
cp : X - A^XA, X, A
6
C AA 2 = 7I, 7

K ([16], p . 248) . Thu s M p = M .
But M p = $p () E(L A) = ^ n E(L A).
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