TheK-admissibility of 2A 6 and 2A 7

LetK be a field and letG be a finite group.G isK-admissible if there exists a Galois extensionL ofK withG=Gal(L/K) such thatL is a maximal subfield of a centralK-division algebra. This paper contains a characterization of those number fields which areQ ₁₆-admissible. This is the same class of number fields which are 2A ₆=SL(2, 9) and 2A ₇ admissible. Dedicated to John Thompson to celebrate his Wolf Prize in Mathematics 1992     

Q-admissibility of certain nonsolvable groups

A finite groupG is calledQ-admissible if there exists a finite dimensional central division algebra overQ, containing a maximal subfield which is a Galois extension ofQ with Galois group isomorphic toG. It is proved thatS ₅ ⁻ , one of the two nontrivial central extensions ofS ₅ byZ/2Z, isQ-admissible. As a consequence of that result and previous results of Sonn and Stern, every finite Sylow-metacyclic group, havingA ₅ as a composition factor, isQ-admissible. This paper is part of a M.Sc. thesis written at the Technion — Israel Institute of Technology, under the supervision of Professor J. Sonn, whom the author wishes to thank for his valuable guidance.     

Sylow-metacyclic groups andQ-admissibility

A finite groupG isQ-admissible if there exists a division algebra finite dimensional and central overQ which is a crossed product forG. AQ-admissible group is necessarily Sylow-metacyclic (all its Sylow subgroups are metacyclic). By means of an investigation into the structure of Sylow-metacyclic groups, the inverse problem (is every Sylow-metacyclic groupQ-admissible?) is essentially reduced to groups of order 2 ^a 3 ^b and to a list of known “almost simple” groups.     

The K-admissibility of SL(2, 5)

Let K be a field and let G be a finite group. G is K-admissible if there exists a Galois extension L of K with G=Gal(L/K) such that L is a maximal subfield of a central K-division algebra. We characterize those number fields K such that H is K-admissible where H is any subgroup of SL(2, 5) which contains a S ₂-group. The method also yields refinements and alternate proofs of some known results including the fact that A ₅ is K-admissible for every number field K.Dedicated to Professor Jacques Tits on the occasion of his sixtieth birthdayThe first author was partly supported by NSF fellowship DMS-8601130; the second author was partly supported by NSF grant DMS-8806371.     

Admissibility and field relations

Let K be a number field. A finite group G is called K-admissible if there exists a G-crossed product K-division algebra. K-admissibility has a necessary condition called K-preadmissibility that is known to be sufficient in many cases. It is a 20-year-old open problem to determine whether two number fields K and L with different degrees over ℚ can have the same admissible groups. We construct infinitely many pairs of number fields (K,L) such that K is a proper subfield of L, and K and L have the same preadmissible groups. This provides evidence for a negative answer to the problem. In particular, it follows from the construction that K and L have the same odd order admissible groups.     

On the admissibility of finite groups over global fields of finite characteristic

Let k be a global field of characteristic p. A finite group G is called k-admissible if there exists a division algebra finite dimensional and central over k which is a crossed product for G. Let G be a finite group with normal Sylow p-subgroup P. If the factor group G/P is k-admissible, then G is k-admissible. A necessary condition is given for a group to be k-admissible: if a finite group G is k-admissible, then every Sylow l-subgroup of G for l ≠ p is metacyclic with some additional restriction. Then it is proved that a metacyclic group G generated by x and y is k-admissible if some relation between x and y is satisfied.     

Finite-to-finite universal quasivarieties are Q-universal

Let K be a quasivariety of algebraic systems of finite type. K is said to be universal if the category G of all directed graphs is isomorphic to a full subcategory of K. If an embedding of G may be effected by a functor F:G K which assigns a finite algebraic system to each finite graph, then K is said to be finite-to-finite universal. K is said to be Q-universal if, for any quasivariety M of finite type, L(M) is a homomorphic image of a sublattice of L(K), where L(M) and L(K) are the lattices of quasivarieties contained in M and K, respectively.?We establish a connection between these two, apparently unrelated, notions by showing that if K is finite-to-finite universal, then K is Q-universal. Using this connection a number of quasivarieties are shown to be Q-universal. Received February 8, 2000; accepted in final form December 23, 2000.     

A characterization of a class of [Z] groups via Korovkin theory

We characterize all the central topological groupsG for which the centreZ(L ¹(G)) of the group algebra admits a finite universal Korovkin set. It is proved thatZ(L ¹(G)) has a finite universal Korovkin set iffĜ is a finite dimensional, separable metric space. This is equivalent to the fact thatG is separable, metrizable andG/K has finite torsion free rank, whereK is a compact open normal subgroup of certain direct summand ofG.     

On Tate-Shafarevich groups over galois extensions

LetA be an abelian variety defined over a number fieldK. LetL be a finite Galois extension ofK with Galois groupG and let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups ofA overK and ofA overL. Assuming these groups are finite, we compute [III(A/L) ^G ]/[III(A/K)] and [III(A/K)]/[N(III(A/L))], where [X] is the order of a finite abelian groupX. Especially, whenL is a quadratic extension ofK, we derive a simple formula relating [III(A/L)], [III(A/K)], and [III(A ^x/K)] whereA x is the twist ofA by the non-trivial characterχ ofG.     

Automorphisms of finite groups of bounded rank

LetG be a finite group admitting an automorphismα withm fixed points. Suppose every subgroup ofG isr-generated. It is shown that (1)G has a characteristic soluble subgroupH whose index is bounded in terms ofm andr, and (2) if the orders ofα andG are coprime, then the derived length ofH is also bounded in terms ofm andr. To Professor John Thompson, in honor of his outstanding achievements     

On existence of finite universal Korovkin sets in the centre of group algebras

In this paper, we characterize compact groupsG as well as connected central topological groupsG for which the centreZ(L ¹(G)) admits a finite universal Korovkin set. Also we prove that ifG is a non-connected central topological group which has a compact open normal subgroupK such thatG=KZ, thenZ(L ¹(G)) admits a finite universal Korovkin set if is a finite-dimensional separable metric space or equivalentlyG is separable metrizable andG/K has finite torsion-free rank.     

A note on a theorem of W. Gaschütz and N. Itô

Given a finite groupG andp an odd prime number, we conclude thatO ^p(G)G isp-nilpotent when for every subgroupH ofG of orderp there exists a subgroupK ofG such thatG=HK andH permutes with every subgroup ofK.     

Complete precones on noncommutative integral domains

A subgroup H is called Q-supplemented in a finite group G, if there exists a subgroup K of G such that G = HK and H ∩ K is contained in H _QG , where H _QG is the maximal quasinormal subgroup of G contained in H. In this article, we investigate the influence of Q-supplementation of some primary subgroups in finite groups. Some recent results are generalized.     

Universal finite group extensions and a non-splitting theorem

LetG andK be finite groups whose orders have a common prime divisor. Then there is a groupK ^* closely related toK for which there is a non-split extension ofK ^* byG. I wish to express thanks to the Mathematics Institute of the Hebrew University of Jerusalem for its hospitality from September to December 1972, and to Dr. Avinoam Mann for his helpful comments.     

Groups containing a strongly embedded subgroup

An involution v of a group G is said to be finite (in G) if vv ^g has finite order for any g ∈ G. A subgroup B of G is called a strongly embedded (in G) subgroup if B and G\B contain involutions, but B ∩ B ^g does not, for any g ∈ G\B. We prove the following results. Let a group G contain a finite involution and an involution whose centralizer in G is periodic. If every finite subgroup of G of even order is contained in a simple subgroup isomorphic, for some m, to L ₂(2 ^m ) or Sz(2 ^m ), then G is isomorphic to L ₂(Q) or Sz(Q) for some locally finite field Q of characteristic two. In particular, G is locally finite (Thm. 1). Let a group G contain a finite involution and a strongly embedded subgroup. If the centralizer of some involution in G is a 2-group, and every finite subgroup of even order in G is contained in a finite non-Abelian simple subgroup of G, then G is isomorphic to L ₂(Q) or Sz(Q) for some locally finite field Q of characteristic two (Thm. 2). Supported by RFBR (project No. 08-01-00322), by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-334.2008.1), and by the Russian Ministry of Education through the Analytical Departmental Target Program (ADTP) “Development of Scientific Potential of the Higher School of Learning” (project Nos. 2.1.1.419 and 2.1.1./3023). (D. V. Lytkina and V. D. Mazurov) Translated from Algebra i Logika, Vol. 48, No. 2, pp. 190–202, March–April, 2009.     

Cusp forms

LetG andH ⊂G be two real semisimple groups defined overQ. Assume thatH is the group of points fixed by an involution ofG. Letπ ⊂L ²(H\G) be an irreducible representation ofG and letf επ be aK-finite function. Let Γ be an arithmetic subgroup ofG. The Poincaré seriesP _f(g)=Σ_H∩ΓΓ f(γ{}itg) is an automorphic form on Γ\G. We show thatP _f is cuspidal in some cases, whenH ∩Γ\H is compact. Partially supported by NSF Grant # DMS 9103608.     

X-inner automorphisms of crossed products and semi-invariants of Hopf algebras

LetR*G be a crossed product of the groupG over the prime ringR and assume thatR*G is also prime. In this paper we study unitsq in the Martindale ring of quotientsQ ₀(R*G) which normalize bothR and the group of trivial units ofR*G. We obtain quite detailed information on their structure. We then study the group ofX-inner automorphisms ofR*G induced by such elements. We show in fact that this group is fairly close to the group of automorphisms ofR*G induced by certain trivial units inQ ₀(R)*G. As an application we specialize to the case whereR=U(L) is the enveloping algebra of a Lie algebraL. Here we study the semi-invariants forL andG which are contained inQ ₀(R*G) and we obtain results which extend known properties ofU(L). Finally, every cocommutative Hopf algebraH over an algebraically closed field of characteristic 0 is of the formH=U(L)*G. Thus we also obtain information on the semi-invariants forH contained inQ ₀(H). Research supported in part by N.S.F. Grant Nos. MCS 83-01393 and MCS 82-19678.     

The admissibility of M11 over number fields

A group G is $\mathbb{Q}$-admissible if there exists a G-crossed product division algebra over $\mathbb{Q}$. The $\mathbb{Q}$-admissibility conjecture asserts that every group with metacyclic Sylow subgroups is $\mathbb{Q}$-admissible. We prove that the Mathieu group $M_{11}$ is $\mathbb{Q}$-admissible, in contrast to any other sporadic group.     

Locally averaged risk

Summary A heuristic method of reducing a class of admissible or Bayes decision rules is given. A new risk function is defined which is called the locally averaged risk. Bayes and admissible rules with respect to the new risk function are calledG-Bayes andG-admissible, respectively. It is shown under general assumptions that the class ofG-Bayes decision rules is a subset of the class of Bayes decision rules and the class ofG-admissible decision rules is a subset of the class of admissible decision rules. Some examples are considered, showing that the usual estimates of the parameter of a distribution with squared error as loss function, which are known to be admissible, are alsoG-admissible. This work was supported in part by NASA Grant-NGR 15-003-064 and NSF Grant-GP 7496 at Indiana University.     

Amitsur cohomology of cubic extensions of algebraic integers

LetK be the rational fieldQ or a complex quadratic number field other than . LetL be a normal three-dimensional field extension onK. IfR andS are the rings of algebraic integers ofK andL respectively, then the Amitsur cohomology groupH ² (S/R, U) is trivial. Inflation and class numbers give information about cohomology arising from certain nonnormal cubic extensions. This work was supported in part by NSF Grant GP-28409.