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Let M be a vector space of dimension n over a field K of characteristic ≠2 and f a non-degenerate quadratic form on M. The automorphisms of the orthogonal group On(K,f), the rotation group On+(K,f), and their corresponding projective groups POn(K,f), POn+(K,f), were determined by J. Dieudonné for n sufficiently large under the condition that the index of f is greater than zero (see 2).

Let M be a finite-dimensional C3 manifold supplied with a C2 Finsler metric ds = F(x, dx), which is not necessarily even in dx. Let p designate the induced oriented topological metric. For any p E M, the antipodal locus of p is the set A(p)= {qEMIp(p, q)_p(p, r) for all rEM}. For example, if M is a real projective space with the Riemannian metric of constant curvature 1, A (p) is a smooth hypersurface (in fact, a projective hyperplane) for every pEHM. One may ask, how close does this property come to characterizing real projective spaces among Finsler manifolds? We prove

Let a finite group of order 2k act freely on an n-sphere. Then there is a generic immersion of an (n 1)-dimensional projective space in the quotient space; the induced homomorphism on the fundamental group is injective. Any such immersion has at least (k) I n-tuple points. A better lower bound is also given. In [1] an answer to a question that was posed to me by John Wood was presented. Specifically, any generic substantial immersion of a projective plane in a 3-dimensional lens space of type (2k, q) must have at least (k) I triple points. The appropriate generalization of that result is contained herein. Suppose that a finite group C/, of order 2k acts freely and properly discontinuously on Sn . Then Milnor's condition [2] gives a unique element of order 2 that is necessarily central. Let G denote the quotient group Gd/(Z/2). Let G() ffgl, *g ,}C G: gi :A gjl} Then G acts on {(G)} as follows: {g1 '.. gr}g :{g g . grg}. Let B(G, r) denote the number of orbits under this action. All maps are smooth; all immersions are to be put into general position. The n-tuple point manifold of such an immersion is 0-dimensional and thus forms a discrete set. The main result can now be stated. Theorem 1. Any generic immersion f: RPn-2 __ Snl/0 such that 7rI(f) # 0 has at least B(G, n) n-tuple points. The number of orbits, B(G, r), of the group G acting on its r-fold subsets is computed below in Theorem 3. Proof. The induced homomorphism, 7 (f), on the fundamental group is an injection. Thus there is an element of order 2 in C; this involution must Received by the editors March 28, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 57S25; Secondary 57R42, 45,95. ? 1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page

Let V V be a finite dimensional unitary space and V = V 1 + ⋯ + V k V = {V_1} + \cdots + {V_k} a direct decomposition. Let P i {P_i} be the orthogonal projector in V V with range V i {V_i} . If A = P 1 + ⋯ + P k A = {P_1} + \cdots + {P_k} we prove that 0 > det ( A ) ≦ 1 0 > \det (A) \leqq 1 and det ( A ) = 1 \det (A) = 1 if and only if V = V 1 + ⋯ + V k V = {V_1} + \cdots + {V_k} is an orthogonal decomposition. Let N i ( 1 ≦ i ≦ k ) {N_i}(1 \leqq i \leqq k) be a normal operator in V V , of rank r i {r_i} . Assume that N = ∑ v = 1 k N i N = \sum _{v = 1}^k{N_i} has rank r = r 1 + ⋯ + r k ≦ n = dim ⁡ V r = {r_1} + \cdots + {r_k} \leqq n = \dim V . If the nonzero eigenvalues of N N (counting multiplicities) are the same as the nonzero eigenvalues of all N i ( 1 ≦ i ≦ k ) {N_i}(1 \leqq i \leqq k) together, then N i N j = 0 {N_i}{N_j} = 0 for i ≠ j i \ne j . This generalizes a recent result of L. Brand.

N. B. BACKHOUSEAbstract. If the locally compact abelian group G has a finite-dimensional unitary irreducible projective representation withfactor system co (i.e. G has an a>-rep), then a subgroup G(co) isdefined which fulfils three roles. First, the square-root of the indexof G(u)) in G is the dimension of every co-rep. Secondly, the co-repsof G can be labelled by the dual group of G(co), up to unitaryequivalence. Thirdly, the essential projective form of an co-rep isdetermined by a unique projective representation of the finitegroup G/G(co).

The purpose of this paper is to develop the theory of elementary divisors, to prove the approximation theorem, and to determine the class number for the following two types of algebraic groups: (i) the unitary group of a hermitian form over an algebraic number field; (ii) the unitary group of a hermitian form over a quaternion algebra, having an algebraic number field as center, with respect to the canonical involution. The latter includes the symplectic group as a special case, since we admit the total matric algebra of degree two as a quaternion algebra. Of course, the approximation theorem is obtained only in the case where the hermitian form is indefinite. A problem of the same kind was investigated for the group of regular elements of a simple algebra by Eichler [5], and for the orthogonal group by Eichler [6, 7] and Kneser [9]. In [11], we treated the groups of type (ii) with the same intent and obtained a somewhat weaker result than in the present paper. We now explain our result by giving a summary of each section. Let F be an algebraic number field of finite degree and K a quadratic extension of F. Let V be a vector space over K of dimension n and ?F(x, y) a non-degenerate hermitian form on V with respect to the non-trivial automorphism of K over F. We denote by U( V, F) and G( V, F) respectively the unitary group of iF and the group of similitudes of iF (cf. 1.1 below). Further, we denote by SU(V, (D) the subgroup of U(V, iF) consisting of the elements with determinant 1. If n is even, we can consider the group of direct similitudes H( V, iF) (cf. 2.2). In the first two sections, we give preliminaries for these four groups, and study elementary properties of maximal lattices, whose definition is as follows. Let g and x be the ring of integers in F and in K, respectively. Let L be an x-lattice in V. We denote by Me(L) the ideal in F generated by the elements iF(x, x) for all x e L, and call it the norm of L. We say that L is maximal if L is a maximal one among the x-lattices with the same norm. Let p be a prime ideal of F, and F, the completion of F with respect to P; and let K, = K ?F Fp and V, = V OF Fi. In ?? 3 and 4, we treat the local theory of elementary divisors for the lattices in V,. If p decomposes in F, KP is isomorphic to F, x F,, and therefore things are much easier. Section 3 is concerned with this case. In ? 4, we discuss the case where K, is a field. 369