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The purpose of this article is to prove the conjecture of L. Auslander that every nilpotent group of affine motions of R n {{\mathbf {R}}^n} that is simply transitive on R n {{\mathbf {R}}^n} has a nontrivial translation in its center. The preliminary result that every such group is unipotent is of independent interest.

In this paper we study families of unipotent algebraic groups over integral rings. The main results relate to the geometry of such families. In particular, we prove that, under some hypotheses, the space of such a family is isomorphic to an affine space over the base. We give counterexamples showing that in the case of an arbitrary base ring the basic facts of the theory of unipotent algebraic groups over a field cease to be true. For a certain class of the group schemes that we consider we prove results on cohomology, extensions and deformations.

In this paper, Steinberg’s concept of a regular element in a semisimple algebraic group defined over an algebraically closed field is generalized to the concept of a k k -regular element in a semisimple algebraic group defined over an arbitrary field of characteristic zero. The existence of semisimple and unipotent k k -regular elements in a semisimple algebraic group defined over a field of characteristic zero is proved. The structure of all k k -regular unipotent elements is given. The number of minimal parabolic subgroups containing a k k -regular element is given. The number of conjugacy classes of R R -regular unipotent elements is given, where R R is the real field. The number of conjugacy classes of Q p {Q_p} -regular unipotent elements is shown to be finite, where Q p {Q_p} is the field of p p -adic numbers.

The main theorem gives necessary and sufficient conditions for the rational group algebra QG to be without (nonzero) nilpotent elements if G is a nilpotent or F·C group. For finite groups G, a characterisation of group rings RG over a commutative ring with the same property is given. As an application those nilpotent or F·C groups are characterised which have the group of units U(KG) solvable for certain fields K.

In this paper, complex Lie groups acting transitively and effectively on complex manifolds with solvable (nilpotent) fundamental groups are studied. It is shown that if is nilpotent, then locally , where is semisimple and is nilpotent. In the case when is solvable, the Levi decomposition of the group is direct if and only if the stationary subgroup contains a maximal unipotent subgroup of the semisimple part. The question of the existence of transitive semisimple groups on is considered.

An algebraic group G is called submersive if every quotient in affine schemes c G : Spec A → Spec A G {c^G}:{\text {Spec}}\;A \to {\text {Spec}}\;{A^G} which is surjective is also submersive. We prove that every unipotent group is submersive. Suppose G is submersive. We show that if c G ( Spec A ) {c^G}({\text {Spec}}\;A) is open in Spec A G {\text {Spec}}\;{A^G} or if some restrictions on the action of G on A are made, c G {c^G} is a topological quotient. A criterion for semisimplicity of points is extended to the case where G is unipotent. Finally, applications of the theory are provided.

The isomorphism problem for finite groups of odd order and nilpotency class 2 with cyclic centre will be solved using some results of Brady [1], [2]. Since a finite nilpotent group is the direct product of its Sylow subgroups, we only need to consider finite q-groups where q is a prime. It has been shown in [1] and [2] that a finite q-group of nilpotency class 2 with cyclic centre is a central product either of two-generator subgroups with cyclic centre or of two-generator subgroups with cyclic centre and a cyclic subgroup, and that the q-groups of class 2 on two generators with cyclic centre comprise the following list: , and if q = 2 we have as well .

Conditions are established under which a given irreducible group of matrices over a perfect field ? is an irreducible component of some reducible maximal locally nilpotent group of matrices over ?.

In [3, p. 149], J. Lambek gives a proof of a theorem, essentially due to Grothendieck and Dieudonne, that if R is a commutative ring with 1 then R is isomorphic to the ring of global sections of a sheaf over the prime ideal space of R where a stalk of the sheaf is of the form R/0P, for each prime ideal P, and . In this note we will show, this type of representation of a noncommutative ring is possible if the ring contains no nonzero nilpotent elements.

A matrix A of finite degree f over a field F is said to be unipotent if A−If is nilpotent; the matrix is said to be projectively unipotent if some multiple is unipotent. If χ(F) > 0 then A is unipotent if and only if A is an element of χ(F)-power order of GL(f, F). A new proof of Kolchin's theorem stating that the identity group of degree 1 is the only irreducible linear group of unipotent matrices over F is given. If f > 2 then a nilpotent subgroup G of GL(f, F) for which the (f−1)-st term of the descending central series contains an element the order of which is divisible by χ(F) consists entirely of projectively unipotent matrices and ʒ f−1 (G) consists of transvections. Conversely every transvection is obtainable in this way, using the conjugates of the triangular subgroup with only 1 in the diagonal. A unipotent matrix is characterized as a matrix which is contained in some nilpotent linear group which is generated by transvections. The case f = 2 also is cleared up.

It is well known that if (n, ϕ(n)) = 1, where ϕ(n) denotes the Euler ϕ function, then the only group of order n is the cyclic group. This is a special case of a more general result due to Dickson (2, p. 201); namely, ifwhere the pi are distinct primes and each αi > 0, the necessary and sufficient conditions that the only groups of order n are abelian are (1) each αi ≤ 2 and (2) nois divisible by any p1 … , ps.We wish to establish a theorem which includes these two results. We let G(n) equal the number of groups of order n whereand we seek necessary and sufficient conditions on n so thatClearly, this problem is equivalent to finding necessary and sufficient conditions on n so that all existing groups of order n be nilpotent.

Let(1) be a word in two variables X, Y, i.e. an element in the free group F2 on two generators X, Y. Let us say that f defines an associative composition for a group G if for arbitrary elements a, b, c in G we have(2) where a ° b is defined by(3)

In a recent conversation with F. Haimo the question arose as to whether a nilpotent group always possesses an outer automorphism. The object of this note is to show that the answer is in the affirmative for certain nilpotent groups of class 2 and also to show that if the group is finite but not Abelian, then for all primes p when pk divides the group order it also divides the order of the group of automorphisms.

The structural characteristic of the normal divisor in a locally nilpotent torsion-free group is given. Moreover, a property of structural isomorphisms of locally nilpotent groups containing no less than two independent elements of infinite order is proved: if H is the subgroup of the mentioned group G, N(H) is its normalizer in G, andϕ is a structural isomorphism of the group G, then N(H) ϕ = N(H ϕ ).

We give a classification of irreducible affine algebraic varieties which are quasihomogeneous with respect to a regular action by a connected linear group of automorphisms and are such that the isotropy subgroup of a point in general position contains a maximal unipotent subgroup of the group of transformations. We find criteria for the normality and factoriality of such varieties. We compute the divisor class group and give a complete description of the orbits in such varieties.

We prove that the trajectory of a lattice of complete measure under the action of a one-parameter group of unipotent linear transformations does not tend to infinity in the lattice space. Bibliography: 3 items.

$ 1. INTRODUCTION In 14; Theorem 31, it was shown that every infinite ZPC-group G (this class of groups includes both FC-groups and nilpotent groups) has an abelian subgroup A such that exp IAl > ]Gl. This paper investigates the possibilities of strengthening this result when G is either PC or nilpotent; in fact, in Theorem 3 we show that if one assumes G.C.H., this result can not be strengthened. We accomplish this by exhibiting an infinite group 0, which is both nilpotent and FC, in which every maximal abelian subgroup A has the property that exp lA[ = ICI. 5 2. NOTATION Let G be a group. Z&(G) will denote the c~th member of the ascending central series. The hypercenter of G is the union of all the terms of the ascending central series. G is ZA or hypercentral if it is its own hypercenter. The class of a ZA group is the first 01 such that Za(G) =G. G is nilpotent if it is a ZA group with finite class. G is PC if each element has only finitely many conjugates. Gi is the derived subgroup of G; E is the identity subgroup. If x, y E G, [x, y] = x-iy-ixy. If 8 is a subset of a group (vector space), (8) is the subgroup (subspace) generated by S. Z, 4 and F, will denote the integers, the rationals and the field with p elemenm, respectively. If R is a commutative ring and M and N are R modules, y : M x M --f N is an R bilinear function if for ri, q E R, vi, vj E M and 1, n E Zf. y is a,lternating (symmetric) (trivial) on M if y(v, w) = - y(w, v) and y(v, w) =

The main purpose of this paper is to derive conclusions about the structure of a nilpotent group of algebra automorphisms and, in the case of a Lie algebra, about the influence of this nilpotence on the structure of the algebra. A motivation for this study is a well known theorem due to Kolchin: A unipotent linear group can be triangularized and is thus nilpotent. The converse is manifestly false, but we have (as an immediate consequence of Theorem 2.7)

Let G be a torsion free, finitely generated nilpotent group, and H be a subgroup of finite index of G. Mal'cev proved in [4] that G can be embedded in a connected, simply connected, nilpotent rational Lie group 65, such that G and H are discrete subgroups of 65 and the coset spaces 6/G and 6/H are compact. Nomizu showed in [5] that if L is the associated Lie algebra of 65, then Hn(L, Q+) -Hn(6/G, Q+) where Q+ is the additive group of rational numbers. Let Hn(G, Q+) denote the rational cohomology group of the abstract group G. It follows from Lie group theory that Hn(65/G, Q+)'t-Hn(G, Q+), hence Hn(L, Q+)--Hn(G, Q+), and for the same reason we have Hn(L, Q+) ,-Hn(H, Q+), whence Hn(G, Q+)>Hn(H, Q+). In this paper we shall, however, indicate a purely algebraic approach to this theorem in the following strengthened form

This note extends the concept of the inner automorphism, but here applies only to those finite groups G for which some member of the lower central series is Abelian. In general (e.g. when G is metabelian) the construction yields an endomorphism semigroup, but in the special case where Gis nilpotent (and may therefore, for our present purposes, be considered as a p-group) a group of automorphisms results.

A negative answer is given to a question posed by E. Schenkman and F. Haimo: does every torsion-free nilpotent group have an outer automorphism?

Necessary and sufficient conditions are determined for the matrix representability over a field of zero characteristic of the second nilpotent product of arbitrary matrix groups and of an arbitrary nilpotent product of Abelian matrix groups.

Let F ( G ) \mathcal {F}(G) denote the set of isomorphism classes of finite homomorphic images of a group G G . We say that groups G G and H H have isomorphic finite quotients if F ( G ) = F ( H ) \mathcal {F}(G) = \mathcal {F}(H) . In this paper we show that if G G is a finitely generated nilpotent group, the finitely generated nilpotent groups H H for which F ( G ) = F ( H ) \mathcal {F}(G) = \mathcal {F}(H) ) lie in only finitely many isomorphism classes. This is done using some finiteness results from the theory of algebraic groups along with some heretofore unpublished results of A. Borel.

(1) It is held that the apparent irreconcilability between physiology of development and genetics does not mean that they actually are separate. They are regarded as two aspects of development and the data of both are accepted. However, the now more prevailing theory of embryonic segregation, and with it all theories of embryonic segregation, and the gene-theory of heredity are rejected as explanations for the physiology of development. (2) Instead of any theory of a segregation, a theory of genetic restriction is proposed, in which it is postulated that the egg from its fertilizable condition, when it has capacity for the production of many embryos, is steadily restricted in its potencies. From a condition of pluripotency it becomes unipotent; then the blastomeres of this unipotent system with first cleavage or thereafter, depending upon the species of egg, become farther restricted and now give rise only to certain areas of the embryo. The loss of potencies is correlated with the increase of nuclear mat...

In a study [6] of the equivalence of forms, Jordan derived the finiteness of the orthogonal group of a form defined over the complex number field whose degree is at least three and whose discriminant is not zero. Herein we demonstrate the validity of Jordan’s result for any field, with the usual restriction that the characteristic of the field, if not zero, dominate the degree of the form. Jordan’s proof rests on an ingenious inequality argument applied to the coefficients of the form. Our argument begins with a proposition of Bott and Tate [3] which focuses attention on unipotent isometries. The proof provided here (due to Harrison [Sj) is regardless of the characteristic of the underlying field, as is the subsequent elimination of unipotent isometries. The argument is completed by showing the group under consideration to be periodic of bounded period with exponent relatively prime to the characteristic of the field, if it is nonzero. Finiteness then follows from a theorem of Burnside. Rather than a theorem of forms we adopt throughout the equivalent setting of symmetric multilinear maps on a vector space in the hope of achieving greater clarity of exposition and with the risk of obscuring the elementary nature of the argument. Let 0 denote a symmetric multilinear map of degree Y 2 3 on a vector space V of dimension n over a field F whose characteristic p, if nonzero, is greater than r. Let 0 = o(8) denote the orthogonal group of 0 in GL(n,F). Employing the standard extension of the scalar field to its algebraic closure, we may assume that F is algebraically closed. Recall that B is singular if and only if for some 0 # z E V, 0(X,..., z, v) = 0 for all 0 E V.

Colloq. Algebraic Topology, 1962, pp. 104-113, Matematisk Institut, Aarhus Universitet, Denmark. 4. M. F. Atiyah, Thom complexes, Proc. London Math. Soc. (3) 11 (1961), 291310. 5. M. F. Atiyah and J. A. Todd, On complex Stiefel manifolds, Proc. Cambridge Philos. Soc. 56 (1960), 342-353. 6. Sze-Tsen Hu, Homotopy theory, Pure and Applied Mathematics VIII, Academic Press, New York and London, 1959. 7. I. M. James, Spaces associated with Stiefel manifolds, Proc. London Math. Soc. (3) 9 (1959), 115-140.

If G is a finite group and P is a group-theoretic property, G will be called P-max-core if for every maximal subgroup M of G, M/MG has property P where MG = ∩ is the core of M in G. In a joint paper with John D. Dixon and A.H. Rhemtulla, we showed that if p is an odd prime and G is (p-nilpotent)-max-core, then G is p-solvable, and then using the techniques of the theory of solvable groups, we characterized nilpotent-max-core groups as finite nilpotent-by-nilpotent groups. The proof of the first result used John G. Thompson's p-nilpotency criterion and hence required p > 2. In this paper I show that supersolvable-max-core groups (and hence (2-nilpotent)-max-core groups) need not be 2-solvable (that is, solvable). Also I generalize the second result, among others, and characterize (p-nilpotent)-max-core groups (for p an odd prime) as finite nilpotent-by-(p-nilpotent) groups.

The head and foot ofPelmatohydra oligactis function in regeneration as unipotent systems. The exact extent of the unipotent tissue immediately posterior to the tentacles and anterior to the foot is not determined in these experiments. The intermediate tissue is pluripotent.