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For positive integers u u and v v , let L u = [ 1 a m p ; 0 u a m p ; 1 ] L_u=\left [\begin {smallmatrix} 1 & 0 \\ u & 1 \end {smallmatrix}\right ] and R v = [ 1 a m p ; v 0 a m p ; 1 ] R_v=\left [\begin {smallmatrix} 1 & v \\ 0 & 1 \end {smallmatrix}\right ] . Let S u , v S_{u,v} be the monoid generated by L u L_u and R v R_v , and let G u , v G_{u,v} be the group generated by L u L_u and R v R_v . In this paper we expand on a characterization of matrices M = [ a a m p ; b c a m p ; d ] M=\left [\begin {smallmatrix}a & b \\c & d\end {smallmatrix}\right ] in S k , k S_{k,k} and G k , k G_{k,k} when k ≥ 2 k\geq 2 given by Esbelin and Gutan to S u , v S_{u,v} when u , v ≥ 2 u,v\geq 2 and G u , v G_{u,v} when u , v ≥ 3 u,v\geq 3 . We give a simple algorithmic way of determining if M M is in G u , v G_{u,v} using a recursive function and the short continued fraction representation of b / d b/d .

Extending results of a number of authors, we prove that if U U is the unipotent radical of an R \mathbb {R} -split solvable epimorphic subgroup of a real algebraic group G G which is generated by unipotents, then the action of U U on G / Γ G/\Gamma is uniquely ergodic for every cocompact lattice Γ \Gamma in G G . This gives examples of uniquely ergodic and minimal two-dimensional flows on homogeneous spaces of arbitrarily high dimension. Our main tools are the Ratner classification of ergodic invariant measures for the action of a unipotent subgroup on a homogeneous space, and a simple lemma (the ‘Cone Lemma’) about representations of epimorphic subgroups.

Given a group G G , we construct, in a canonical way, an inverse semigroup S ( G ) \mathcal {S}(G) associated to G G . The actions of S ( G ) \mathcal {S}(G) are shown to be in one-to-one correspondence with the partial actions of G G , both in the case of actions on a set, and that of actions as operators on a Hilbert space. In other words, G G and S ( G ) \mathcal {S}(G) have the same representation theory. We show that S ( G ) \mathcal S(G) governs the subsemigroup of all closed linear subspaces of a G G -graded C ∗ {C}^* -algebra, generated by the grading subspaces. In the special case of finite groups, the maximum number of such subspaces is computed. A “partial” version of the group C ∗ { C}^* -algebra of a discrete group is introduced. While the usual group C ∗ { C}^* -algebra of finite commutative groups forgets everything but the order of the group, we show that the partial group C ∗ { C}^* -algebra of the two commutative groups of order four, namely Z / 4 Z Z/4 Z and Z / 2 Z ⊕ Z / 2 Z Z/2 Z \oplus Z/2 Z , are not isomorphic.

We give explicit formulas for representations of classical matrix groups whose Chern characters have lowest order terms equal to standard characteristic classes. For SO(2r), the Euler class e does not arise in this way, but 2r-le does arise in this way. A complex representation p of a compact Lie group G gives rise to a complex vector bundle (p over the classifying space BG of G. In this way one obtains a ring homomorphism R(G) -* K(BG) from the complex representation ring of G to the complex K-theory of BG. Composing with the Chern character yields a ring homomorphism, R(G) -* K(BG) -h H**(BG;Q) Our interest here is in determining which cohomology classes on BG occur as the lowest order terms of some Chern character ch((p). We will consider the classical matrix groups U(n), SU(n), Sp(n), and SO(n), and we will produce explicit representations p whose Chern characters ch(Qp) have lowest order terms equaling standard characteristic classes. In all cases considered here, the lowest order terms of the Chern characters ch((p) turn out to be integral classes. For U(n), SU(n), Sp(n), and SO(2r + 1), every homogeneous integral cohomology class modulo torsion on BG arises as the lowest order term of some Chern character ch((p); for SO(2r), the Euler class does not so arise, but 2r-1 times the Euler class does. In the cases considered here where BG has integral cohomology which is free of torsion, namely, U(n), SU(n), and Sp(n), general results of Atiyah and Hirzebruch ([3], ?2.5 and ?4.8) imply that the lowest order terms of any Chern character ch((p) are integral classes and that any homogeneous integral class arises in this way. Thus our only contribution in these cases is the explicit form of certain representations which give rise to standard characteristic classes. For SO(n), whose classifying space has 2-torsion, the general results of Atiyah and Hirzebruch do not apply directly. It may be possible to obtain some general results in the presence of torsion by using the Atiyah-Hirzebruch spectral sequence together with integrality results such as those of Adams [1] and Maunder [7]; we have Received by the editors October 1, 1996. 1991 Mathematics Subject Classification. Primary 55R40. The author was partially supported by NSA grants MDA904-94-H-2025 and MDA904-96-10067, and by Purdue University Calumet Scholarly Research Awards. ?1998 American Mathematical Society 1237 This content downloaded from 207.46.13.172 on Fri, 07 Oct 2016 06:27:24 UTC All use subject to http://about.jstor.org/terms

It is proved that every torsion-free nilpotent group of finite Morley rank is isomorphic to a matrix group over a field of characteristic zero.

We consider the Brauer group ${\rm BM}'(k,G)$ of a group $G$ (finite or infinite) over a commutative ring $k$ with identity. A split exact sequence $$1\longrightarrow {\rm Br}'(k)\longrightarrow {\rm BM}'(k,G)\longrightarrow {\rm Gal}(k,G) \longrightarrow 1$$ is obtained. This generalizes the Fr\"ohlich-Wall exact sequence from the case of a field to the case of a commutative ring, and generalizes the Picco-Platzeck exact sequence from the finite case of $G$ to the infinite case of $G$. Here ${\rm Br}'(k)$ is the Brauer-Taylor group of Azumaya algebras (not necessarily with unit). The method developed in this paper might provide a key to computing the equivariant Brauer group of an infinite quantum group., Comment: 13 pages

Let G be a coproduct of p-primary abelian groups with each factor of cardinality not exceeding , , and let F be a perfect field of characteristic p . If V(G) is the group of normalized units of the group algebra F(G), it is shown that G is a direct factor of V(G) and that the complementary factor is simply presented. This generalizes a theorem of W. May, who proved the result in the case when G itself has cardinality not exceeding N, and length not exceeding co I Throughout let F denote a perfect field of characteristic p t 0 and let G denote a p-primary abelian group. In a recent paper, W. May [M2] proved the following result, where V(G) denotes the group of normalized units in the group algebra F(G); hence V(G) = { cigi E F(G): Ec = 1}. Theorem 1 ([M2]). If IGI < t1 and the length of G does not exceed wo1, then G is a direct factor of V(G) and the complementary factor is simply presented. Recall that a group is simply presented if it can be presented by generators and relations that involve at most two generators. The main purpose of this note is to show that the condition on the length of G in the above theorem can be omitted. In other words, the theorem is proved for a much more general class of groups (inasmuch as there exists an abundance of abelian p-groups of cardinality not exceeding t1 with arbitrary prescribed length i < wd2 ). Actually, we extend the theorem to any abelian p-group G which is a coproduct of groups with the cardinality of each factor not exceeding tj; of course such a G can have arbitrarily large cardinality. We refer to [MI] and [M3] as examples of the connection between the direct factor theorem and the isomorphism problem: When does F(G) F(G') imply that G G' ? As is customary for group algebras, we use the multiplicative notation for G even though G is abelian. Consequently, for an ordinal a, G' is the translation of the more familiar p7G; since there is only one relevant prime, Received by the editors November 28, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 20C)7; Secondary 20K10. Supported in part by NSF grant DMS 8800862. ? 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page

Given a complex representation of a finite group G G , construct the integer valued function α \alpha on G G by setting α ( g ) \alpha (g) to be the dimension of the fixed-point-space of g g in the module corresponding to the given representation. Usually, α \alpha is not a generalized character of G G and for trivial reasons | G | α |G|\alpha is always a generalized character. The main result of this paper is that e α e\alpha is always a generalized character, where e e is the exponent of G G .

Matrices for which the group inverse exists and is nonnegative are studied. Such matrices are characterized in terms of a generalization of monotonicity. In particular, nonnegative matrices with this property are characterized in terms of their nonnegative rank factorizations.

It is the purpose of this paper to find in terms of parameters the most general matrix X which is K-commutative with respect to a given matrix A. The proofs will yield a method of rational construction for such a matrix X.

A permanent group is a group of nonsingular matrices on which the permanent function is multiplicative. We consider only permanent groups which contain the group of nonsingular diagonal matrices. If the underlying field is infinite of characteristic zero or greater than n, then each such permanent group consists only of matrices in which exactly one diagonal has all nonzero entries. A permanent group is a group of nonsingular matrices on which the permanent function is multiplicative. One example is 3>„(F), the group of nonsingular n X« diagonal matrices over the field F. Marcus and Mine [3] conjectured that A„, the groups of «xn matrices of the form PD where P is a permutation matrix and D e 9n(F) is a maximal permanent group. In this conjecture the field F was not specified and the first author [1] verified the conjecture for the field of complex numbers. In this paper we consider the set 3^n(F) of those permanent groups which contain @n(F), and characterize ^n(F) when Fis an infinite field with char F—0 or >«. In [2] we defined the sets/n(F) of all« x« matrix groups G of nonsingular matrices over F satisfying: (i) 2?>n(F) „(F) where "»" denotes the Hadamard product. Here, we are also concerned with the set ¥i n(F) of «x« nonsingular matrix groups over F such that G = H ■ K={hk, h e H,he K} where: (i) H es/„(F), (ii) A" is a group of «xn permutation matrices, and (iii) (PHP-l\PeK)esfn(F). Note that 2JJF)^H for every Hes/n(F) so that 2n(F)^G for every G e

Introduction. L. Bieberbach in two fundamental papers [2], [3] established the fundamental theorems for the crystallographic groups or Raumgruppen. We propose in this paper to give an almost completely self-contained account of these fundamental facts. We will use only the elementary theory of groups, matrices and polynomials from algebra, the basic geometry of euclidean space and the most elementary topological considerations. At one point we will need the exponential mapping for Lie matrix groups for which various elementary accounts are available. I would like to thank M. Rosenlicht and P. Fong for useful conversations.

1. M. Heins, Algebraic structure and conformal mapping, Technical Report issued by the Institute for Advanced Study, 1957. 2. H. L. Royden, Rings of analytic and meromorphic functions, Trans. Amer. Math. Soc. vol. 83 (1956) pp. 269-276. 3. W. Rudin, Some theorems on bounded analytic functions, Trans. Amer. Math. Soc. vol. 78 (1955) pp. 333-342. 4. H. Weyl, Die Idee der Riemannschen Flaechen, Berlin, 1923.

Classical theorems of Schur and Burnside on finiteness of matrix groups are generalized to regular and inverse matrix semigroups. A group G is periodic if every element of G has finite order. G is periodic of boundedperiod iff there is a bound on the orders of all elements of G. The following classical theorems give conditions for finiteness of periodic groups of matrices: BURNSIDE'S THEOREM [3, (36.1)]. A periodic group of complex n x n matrices of bounded periodic is finite. SCHUR'S THEOREM [3, (36.2)]. A finitely generated periodic group of complex n x n matrices is finite. In [4], we have observed that both theorems generalize trivially to irreducible matrix semigroups and that Burnside's theorem is false in general for semigroups. In this note we prove that Schur's theorem generalizes to regular semigroups and, as a corollary, that Burnside's theorem generalizes to inverse semigroups. Recall that for elements x, y of a semigroup, y is a generalized inverse of x if xyx=x and yxy=y. A semigroup S is a regular (inverse) if every element of S has a (unique)l generalized inverse. We will need the following theorem. THEOREM (COUDRAIN-SCHUTZENBERGER [2]). Let S be a finitely generated monoid. Assume (1) S satisfies the descending chain condition on principal two-sided ideals. (2) For all x, y in S, {x, y} ' Sx nyS implies {x, y} ' xS nSy. (3) All subgroups of S are finite. Then S is finite. Received by the editors May 23, 1972. AMS (MOS) subject classifications (1970). Primary 20M30.

Let GL(n, R) denote the real general linear group with the usual topology. A subgroup of GL(n, R) will be called an w-matrix group. We will say that a subgroup G of GL(n, R) is a discrete group if it is discrete in the induced topology. If for any group G, [G, G] denotes the commutator subgroup of G, we will say that a group 5 is solvable provided the sequence of groups S = Slt [Si, Si] = S2, • • • , [Sic-i, Sk-i]=Sk, terminates in the identity element for some k. If Sk = e and Sk-i^e, where e is the identity in S, then we will say that S is k step solvable and k is called also the index of solvability. Then in [l] and [4] we have independent proofs of the following theorem