We shall determine in this paper groups of types Dn, E6, E7 and E8 generated by SU(3, q)'s, q odd, q > 3. These groups are defined in Phan (1975). [We shall refer to this paper as I]. Acquaintance with the results of I is assumed. The identification of groups of type D4 is similar to that of SU(n, q). We actually construct an isomorphism from the universal group of type D4 onto Spin+(8, q). This direct approach does not appear to be feasible for groups of type Dn with n ≧ 5. Fortunately Wong's recent result (1974) is applicable here. But his theorem requires that the characteristic of the field be odd; hence unlike the unitary case, we assume that q is odd and q 3. Using Wong's theorem, we proceed to show by induction that groups of type Dn are homomorphic images of Spin+(2n, q) or Spin −(2n, q) according as n is even or n is odd.

If C is a group which satisfies the maximal condition for normal subgroups, then C may be cancelled from a group A in direct products if and only if the infinite cyclic group can be cancelled from A. Finitely generated torsion free nilpotent groups of class 2 satisfy a Remak Krull Schmidt condition.

During the last two decades a variety of methods have been developed for the problem of estimation of unknown density f wrt Lebesgue measure and its vth dervative g(= f(v)) using i.i.d random variables X1, …, Xn when X1 ∼ f. For example, see Wegman (1972). In almost all the papers on the estimation of f(x) or g(x), various authors assumed the existence of derivatives of f of order r(>v) at x to obtain rates for the mean-square convergences and other desirable properties for their estimators. Here it is shown that if

then estimators gˇ(x) can be constructed for which E[(gˇ(x)-g(x))2] = O(n−(2a-δ)/(2a+2v+1)) for anygiveng δ > 0. Simiar statements hold for almost sure convergence of gˇ(x). It can also be shown that (gˇ(x1), gˇ(x2)) is asymptotically bivariate normal under certain conditions for x1, ≠ x2. If (Al) is satisfied with a ≦ l, then our estimators have all the desirable properties while other methods are not applicable in this situation since they require differentiability conditions on g. (For example, see Susarla and Kumar (1975) and its references.) Our estimators are defined by using the inversion theorem for some absolutely integrable characteristic functions. The motivation for our estimators is given in O'Bryan and Susarla (1975, 76) and Susarla and O'Bryan (1975).

Two characterizations of semisimple classes of associative and alternative rings (and semigroups with 0) are given:

(i) A class is a semisimple class if and only if it is hereditary, closed under extensions and subdirect sums;

(ii) A class is a semisimple class if and only if it is hereditary, closed under extensions, and has the co-inductive property.

The first characterization sharpens Armendariz's (1968) result proved for associative rings, the second one is categorically dual to a characterization of radical classes due to Amitsur (1954).

Let p and q be infinite cardinal numbers, p ≧ q, X a set of cardinality p, and BL(X, p, q) the Baer-Levi semigroup of type (p, q) on X. Subsemigroups of BL(X, p, q) are defined and called bounded Baer-Levi semigroups. These semigroups are right simple and are universal in the embedding sense for the class of idempotent-free, right cancellative semigroups S so that any two elements of S have a common right identity in S. Further properties of bounded Baer-Levi semigroups are given and the structure of their lattices of congruences is discussed.

We introduce a class of inverse semigroups of injective transformations and our main result concerns isomorphisms between two such semigroups. This result is then applied to semigroups of homeomorphisms between closed subsets of a T1 topological space, semigroups of homeomorphisms between compact subsets of k-spaces and semigroups of isomorphisms between subsemilattices of semilattices. In the first two cases it is shown that the two inverse semigroups under consideration are isomorphic if and only if the corresponding topological spaces are homeomorphic. In the latter case, the two inverse semigroups are isomorphic if and only if the semilattices are either isomorphic or are dual isomorphic infinite chains.

A doubly transitive permutation group of degree p2 + 1, p a prime, is proved to be doubly primitive for p ≠ 2. We also show that if such a group is not triply transitive then either it is a normal extension of P S L (2, p2) or the stabilizer of a point is a rank 3 group.

By a variety of topological algebras we mean a class V of topological algebras of a fixed type closed under the formation of subalgebras, products and quotients (i.e. images under continuous homomorphisms yielding the quotient topology). In symbols, V = SV = PV = QV. if V is also closed under the formation of arbitrary continuous homomorphic images, then V is a wide variety. variety. As an example we have the full variety V = Modr (Σ), the class of all topological algebras of a fixed type τ obeying a fixed set Σ of algebraic identities. But not every wide variety is full, e.g. the class of all indiscrete topological algebras of a fixed type; in fact, as Morris observed (1970b), there exists a proper class of varieties of topological groups.

We construct an associative tensor product for 2-distributors by means of cartesian coends and we prove the usual adjointness property for a suitable Hom.

Throughout this paper an l-group will always mean an archimedean lattice-ordered group and we shall confine our attention to such groups. An l-group splits if it is a cardinal summand of each l-group that contains it as an l-ideal. Suppose that G is an l-subgroup of an l-group H. Then G is large in H or H is an essential extension of G if for each l-ideal L≠0 of H, L∩G≠0. G is essentially closed if it does not admit any proper essential extension. Conrad (1971) proved that each essentially closed l-group splits, but not conversely.