When are inner mapping groups generated by conjugation maps?

A subloop of a loop Q is said to be normal if it is stabilized by all maps in the inner mapping group of Q. Here we show that in many cases, the inner mapping group of a Moufang loop is actually generated by conjugation maps. This includes any Moufang loop whose cubes generate either the entire loop or a subloop of index three. Such a result can be an extremely useful tool when proving that certain subloops are indeed normal just by showing that they are stabilized by the conjugation maps.     

Loops mit distributiven und geometrischen Unterloopverbänden

The following facts are shown: A loop with a finite distributive subloop lattice is finite, monogenic and all its subloops are monogenic. Therefore, power-associative loops having finite distributive subloop lattices are cyclic groups. A loop G with its subloop lattice L(G) being a finite n-dimensional projective geometry is generated by at most n+1 elements. For all n IN, n4, there are power-associative loops whose subloop lattices are projective lines with n points. Furthermore, for a given projective planeP _n (desarguesian or non-desarguesian) of order n there exists a power-associative loopG with L(G) -P _n.     

On the lattice formed by all normal subloops of a finite loop

All normal subloops of a loopG form a modular latticeL _n (G). It is shown that a finite loopG has a complemented normal subloop lattice if and only ifG is a direct product of simple subloops. In particular,L _n (G) is a Boolean algebra if and only if no two isomorphic factors occurring in a decomposition ofG are abelian groups. The normal subloop lattice of a finite loop is a projective geometry if and only ifG is an elementary abelianp-group for some primep.     

The GL2 Main Conjecture for Elliptic Curves without Complex Multiplication

Let G be a compact p-adic Lie group, with no element of order p, and having a closed normal subgroup H such that G/H is isomorphic to Z_p. We prove the existence of a canonical Ore set S^* of non-zero divisors in the Iwasawa algebra Λ(G) of G, which seems to be particularly relevant for arithmetic applications. Using localization with respect to S^*, we are able to define a characteristic element for every finitely generated Λ(G)-module M which has the property that the quotient of M by its p-primary submodule is finitely generated over the Iwasawa algebra of H. We discuss the evaluation of this characteristic element at Artin representations of G, and its relation to the G-Euler characteristics of the twists of M by such representations. Finally, we illustrate the arithmetic applications of these ideas by formulating a precise version of the main conjecture of Iwasawa theory for an elliptic curve E over Q, without complex multiplication, over the field F generated by the coordinates of all its p-power division points; here p is a prime at least 5 where E has good ordinary reduction, and G is the Galois group of F over Q.     

Commutative Moufang Loops with Finite Conjugacy Classes of Subloops

It is proved that the following conditions are equivalent for an arbitrary commutative Moufang loop :1) the loop is finite over the center;2) every subloop of defines a finite conjugacy class of subloops;3) every associative subloop of defines a finite conjugacy class of subloops;4) every infinite associative subloop of defines a finite conjugacy class of subloops.     

The nilpotence class of the Frattini subgroup

Ap-group of sufficiently large nilpotence class cannot occur as a normal subgroup contained in the Frattini subgroup of any finite group. The Frattini subgroup of a group of order Π _pi ^αi with max α _i at least 3, has nilpotence class at most 1/2 (max α _i − 1). The Frattini subgroup of at-group is abelian. The occurrence of groups of orderp ⁴ as normal subgroups contained in Frattini subgroups is investigated. National Science Foundation Science Faculty Fellow, University of Cincinnati     

Sylow's theorem for Moufang loops

For finite Moufang loops, we prove an analog of the first Sylow theorem giving a criterion for the existence of a p-Sylow subloop. We also find the maximal order of p-subloops in the Moufang loops that do not possess p-Sylow subloops.     

Construction of Modules with Finite Homological Dimensions

A new homological dimension, called G^*-dimension, is defined for every finitely generated module M over a local noetherian ring R. It is modeled on the CI-dimension of Avramov, Gasharov, and Peeva and has parallel properties. In particular, a ring R is Gorenstein if and only if every finitely generated R-module has finite G^*-dimension. The G^*-dimension lies between the CI-dimension and the G-dimension of Auslander and Bridger. This relation belongs to a longer sequence of inequalities, where a strict inequality in any place implies equalities to its right and left. Over general local rings, we construct classes of modules that show that a strict inequality can occur at almost every place in the sequence.     

A note on intersections of free submonoids of a free monoid

According to a theorem of Tilson [6] any intersection of free submonoids of a free monoid is free. Here we consider intersections of the form {x, y}^* ∩ {u, v}^*, where x, y, u and v are words in a finitely generated free monoid Σ^*, and show that if both the monoids {x, y}^* and {u, v}^* are of the rank two, then the intersection is a free monoid generated either by (at most) two words or by a regular language of the form β₀ + β((γ(1+ δ + ... δ^t))^*ε for some words β⁰, β, γ, δ and ε, and some integer t≥0. An example is given showing that the latter possibility may occur for each t≥0 with nonempty values of the words.     

Infinite words and permutation properties

Summary In contrast with the theorem of Restivo and Reutenauer, which in its nontrivial part says that each property P_n (n>-2) ensures the finiteness of a finitely generated periodic semigroup, we prove that no property P _n ^* (n≥3) can do this     

Representing endomorphisms and principal congruences

For every algebraU there is an algebraU ^* with (up to isomorphism) the same endomorphism, subalgebra and congruence structure as that ofU, for which every finitely generated subalgebra and every finitely generated congruence ofU ^* is singly generated. The theorem is proved in a somewhat more general category theoretic context.Presented by R. W. Quackenbush.This author's research was supported by an OTKA grant from Hungary.This author's research was supported by NSERC, The Natural Sciences and Engineering Research Council of Canada.     

Fundamental groups of asymptotic cones

We show that for any metric space M satisfying certain natural conditions, there is a finitely generated group G, an ultrafilter ω, and an isometric embedding ι of M to the asymptotic cone Cone_ω(G) such that the induced homomorphism ι^*:π₁(M)→π₁(Cone_ω(G)) is injective. In particular, we prove that any countable group can be embedded into a fundamental group of an asymptotic cone of a finitely generated group.     

Primary Decompositions in Varieties of Commutative Diassociative Loops

The decomposition theorem for torsion abelian groups holds analogously for torsion commutative diassociative loops. With this theorem in mind, we investigate commutative diassociative loops satisfying the additional condition (trivially satisfied in the abelian group case) that all nth powers are central, for a fixed n. For n = 2, we get precisely commutative C loops. For n = 3, a prominent variety is that of commutative Moufang loops.

Many analogies between commutative C and Moufang loops have been noted in the literature, often obtained by interchanging the role of the primes 2 and 3. We show that the correct encompassing variety for these two classes of loops is the variety of commutative RIF loops. In particular, when Q is a commutative RIF loop: all squares in Q are Moufang elements, all cubes are C elements, Moufang elements of Q form a normal subloop M ₀(Q) such that Q/M ₀(Q) is a C loop of exponent 2 (a Steiner loop), C elements of L form a normal subloop C ₀(Q) such that Q/C ₀(Q) is a Moufang loop of exponent 3. Since squares (resp., cubes) are central in commutative C (resp., Moufang) loops, it follows that Q modulo its center is of exponent 6. Returning to the decomposition theorem, we find that every torsion, commutative RIF loop is a direct product of a C 2-loop, a Moufang 3-loop, and an abelian group with each element of order prime to 6.

We also discuss the definition of Moufang elements and the quasigroups associated with commutative RIF loops.     

On locally graded groups with an Engel condition on infinite subsets

A group G is said to be in E_k^*E_k^* (k a positive integer), if every infinite subset of G contains a pair of elements that generate a k-Engel group.¶It is shown that a finitely generated locally graded group G in E_k^*E_k^* is a finite-by- (k-Engel) group, in particular a finite extension of a k-Engel group.     

Finite extensions of free pro-P groups of rank at most two

For a pro-p groupG, containing a free pro-p open normal subgroup of rank at most 2, a characterization as the fundamental group of a connected graph of cyclic groups of order at mostp, and an explicit list of all such groups with trivial center are given. It is shown that any automorphism of a free pro-p group of rank 2 of coprime finite order is induced by an automorphism of the Frattini factor groupF/F ^* . Finally, a complete list of automorphisms of finite order, up to conjugacy in Aut(F), is given. Supported by an NSERC grant. Supported by the Austrian Science Foundation.     

PI-rings and semiprime rings having faithful modules with krull dimension

It is proved that if a PI-ring R has a faithful left R-module M with Krull dimension, then its prime radical rad(R) is nilpotent. If, moreover, the R-module M and the left ideal_R(rad(R)) are finitely generated, then R has a left Krull dimension equal to the Krull dimension of M. It turns out that a semiprime ring, which has a faithful (left or right) module with Krull dimension, is a finite subdirect product of prime rings. Moreover, first, a right Artinian ring R such that rad(R)²=0 has a faithful Artinian cyclic left module, and second, a finitely generated semiprime PI-algebra over a field has a faithful Artinian module. We give examples showing that the restrictions imposed are essential, as well as an example of a finitely generated prime PI-algebra over a field, which is not Noetherian and has a Krull dimension. Supported by RFFR grant No. 26-93-011-1544. Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 562–572, September–October, 1997.     

Factorization of mappings of topological spaces and homomorphisms of topological groups in accordance with weight and dimension ind

Symbols w(X), nw(X), and hl(X) denote the weight, the network weight, and the hereditary Lindelöf number of a space X, respectively. We prove the following factorization theorems.

1. Let X and Y be Tychonoff spaces, φ: X→Y a continuous mapping, hl(X)≤τ, and w(Y)≤τ. Then there exist a Tychonoff space Z and continuous mappings ψ: X→Z, χ: Z→Y such that φ=χ o ψ, Z=ψ(X), w(Z)≤τ andind Z≤ind X. Moreover, if nw(X)≤τ, then mapping ψ is one-to-one.
2. Let π: G→H be a continuous homomorphism of a Hausdorff topological group G to a Hausdorff topological group H, hl(G)≤τ and w(H)≤τ. Then there are a Hausdorff topological group G^* and continuous homomorphisms g: G→G^*, h: G^*→H so that π=h o g, G^*=g(G), w(G^*)≤τ andind G^*≤ind G. If nw(G)≤τ, then g is one-to-one.
3. For every continuous mapping φ: X→Y of a regular Lindelöf space X to a Tychonoff space Y one can find a Tychonoff space Z and continuous mappings ψ: X→Z, χ: Z→Y such that φ=χ o ψ, Z=ψ(X), w(Z)≤w(Y),dim Z≤dim X, andind ₀ Z≤ind ₀ X, whereind ₀ is the dimension function defined by V.V.Filippov with the help of G_δ-partitions. If we additionally suppose that X has a countable network, then ψ can be chosen to be one-to-one. The analogous result also holds for topological groups.
4. For each continuous homomorphism π: G→H of a Hausdorff Lindelöf Σ-group G (in particular, of a σ-compact group G) to a Hausdorff group H there exist a Hausdorff group G^* and continuous homomorphisms g: G→G^*, h:G^*→H so that π=h o g, G^*=g(G), w(G^*)≤w(H),dimG^*≤dimG, andind G^*≤ind G. Bibliography: 25 titles.

     

Prime ideals in the C*-algebra of a nilpotent group

We say that a locally compact groupG hasT ₁ primitive ideal space if the groupC ^*-algebra,C ^*(G), has the property that every primitive ideal (i.e. kernel of an irreducible representation) is closed in the hull-kernel topology on the space of primitive ideals ofC ^*(G), denoted by PrimG. This means of course that every primitive ideal inC ^*(G) is maximal. Long agoDixmier proved that every connected nilpotent Lie group hasT ₁ primitive ideal space. More recentlyPoguntke showed that discrete nilpotent groups haveT ₁ primitive ideal space and a few month agoCarey andMoran proved the same property for second countable locally compact groups having a compactly generated open normal subgroup. In this note we combine the methods used in [3] with some ideas in [9] and show that for nilpotent locally compact groupsG, having a compactly generated open normal subgroup, closed prime ideals inC ^*(G) are always maximal which implies of course that PrimG isT ₁.