36
M. L. RACINE
must show s - s, = 0 and t - t. = r - r., l j , J? n or s - s = r - r
0
and t - t. = 0, 1 j , o\ n. Assume s
0
- s = s
0
- s = . . . = s, - s. .
£ j * 2 1 3 2 l i - l
and t
0
- t . = r
0
- r_, t - t = r - r
0
, . . . , t. - t. = r. - r. . If
2 1 2 1* 3 2 3 2 ' i
I - I I
l- l
s,^, - s. = r, ,. - r, then t. - t. = 0 (r. ,. - r. 0). Hence s . . . - s_ =
l+l l
I + I I I + I
l l+l
I
l+l 1
r, , - r. and t, ,, - t . = r. - r, . Now one of them is zero, so either
i+n l
I
i+I l i l
s, ,, - s. = r, ,, - r. = 0 in which cas e s. ,, - s, = 0, or t. , _ - t . =
i+l 1 i+l
I
i+l
I
i+l 1
r. - r = 0 so that r. = r = . . . = r. and t
0
- t = t , - t = . . . = t. - t. _ = 0
i l 1 2
I
2 1 3 2
I
i- l
and t - t. = t. - t = 0 . Induction on i yields the desired result.
q . e . d .
PROPOSITION 5. Let * be an involution of & . If F is a maximal
n
^-stabl e order of & , there exist s a maximal order E of i9 with F = E n E*
n n
and if F C E , E a maximal order, then E = E or E*.
PROOF. We know that F C some maximal order E. Therefore
F = F* C E * and by maximality F = E n E*. If F c E E a maximal
order, then the same argument yields F = E n E.#. Clearly E* is maximal
for if not then E* C E maximal and E = E** c E *, a contradiction. By
Lemma 12 E n E* = E nE * implies E = E or E* = E
q. e. d.
Let h be a non-degenerate hermitian or skew-hermitian form on V an
n-dimensional left & -vecto r space ; & act s on V and h induces an involu-
n
tion * of & : A - A* where A* is th e adjoint of A relative to h, so
n
h(xA, y) = h(x, yA*) for all x, y

V. If H is the matrix of h with respect to
a fixed bas e of V, AH = HA* , so A* = HA H . Any involution of & may be
n
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