Arbitrary Subgroup Research Articles

Bounds on Key Appearance Equivocation for Substitution Ciphers

The average conditional entropy of the key given the message and its corresponding cryptogram, ff(K|M, C), which is refer as a key appearance equivocation, was proposed as a theoretical measure of the strength of the cipher system under a known plaintext attack by Dunham in 1980. In the same work (among other things), lower and upper bounds for H(SM |MLCL) are found and its asymptotic behavior as a function of cryptogram length L is described for simple substitution ciphers, i.e., when the key space SLambda1, Is the symmetric group acting on a discrete alphabet M. In the present paper we consider the same problem when the key space is an arbitrary subgroup K < S,M and generalize Dunham's result.

An econometric method of correcting for unit nonresponse bias in surveys

Past approaches to correcting for unit nonresponse in sample surveys by re-weighting the data assume that the problem is ignorable within arbitrary subgroups of the population. Theory and evidence suggest that this assumption is unlikely to hold, and that household characteristics such as income systematically affect survey compliance. The authors show that this leaves a bias in the re-weighted data and they propose a method of correcting for this bias. The geographic structure of nonresponse rates allows them to identify a micro compliance function, which they then use to re-weight the unit-record data. An example is given for the U.S. Current Population Surveys, 1998-2004. The authors find, and correct for, a strong household income effect on response probabilities.

Unions of slender groups

It is well-known that the class of slender groups is closed under extensions, arbitrary direct sums and subgroups. We show that it is also closed under taking unions of continuous well-ordered ascending chains of groups where each factor of consecutive groups is slender (Theorem 2.2). We also prove that the class of slender groups cannot be obtained from any set of slender groups by forming unions of chains of the mentioned kind and taking subgroups (Theorem 4.2). This generalizes a result by Gobel-Wald [11].

Subsimple groups

AbstractWe define a notion of genericity for arbitrary subgroups of groups interpretable in a simple theory, and show that a type generic for such a group is generic for the minimal hyperdefinable supergroup (the definable hull). In particular, at least one generic type of the definable hull is finitely satisfiable in the original subgroup. If the subgroup is a subfield. then the additive and the multiplicative definable hull both have bounded index in the smallest hyperdefinable superfield.

A generalization of Marshall’s equivalence relation

For p prime and for a field F containing a root of unity of order p, we generalize Marshall's equivalence relation on orderings to arbitrary subgroups of F× of index p. The equivalence classes then correspond to free pro-p factors of the maximal pro-p Galois group of F. We generalize to this setting results of Jacob on the maximal pro-2 Galois group of a Pythagorean field.

Automorphisms of Free Left Nilpotent Leibniz Algebras and Fixed Points#

ABSTRACT Let F m (N) be the free left nilpotent (of class two) Leibniz algebra of finite rank m, with m ≥ 2. We show that F m (N) has non-tame automorphisms and, for m ≥ 3, the automorphism group of F m (N) is generated by the tame automorphisms and one more non-tame IA-automorphism. Let F(N) be the free left nilpotent Leibniz algebra of rank greater than 1 and let G be an arbitrary non-trivial finite subgroup of the automorphism group of F(N). We prove that the fixed point subalgebra F(N) G is not finitely generated.

Almost Hamiltonian Groups

In this paper we investigate the following problem in Group Theory: which properties $cal P$ transfer (or do not transfer) from all cyclic subgroups, or all abelian subgroups to all arbitrary subgroups? We solve this problem completely when $cal P$ is the property of having finite index in its normal closure, proving that $cal P$ carries from abelian, but not from cyclic to arbitrary subgroups. We use primarily some results of B.H. Neumann.

On tensor induction for representations of finite groups

It is well known that if D is an irreducible complex representation of a finite group G, then every direct summand of the restriction of D to a subgroup H must have degree at least as large as the degree of D divided by the index | G : H | ; moreover, D is induced from H if and only if the restriction does have a direct summand whose dimension is equal to this quotient. This paper explores the possibility of an analogous result for tensor induction, under the additional assumption that D is faithful, quasi-primitive and not a tensor product (of projective representations of degree greater than 1), and that the Fitting subgroup F ( G ) is not in the centre Z ( G ) . The main question is this: if the restriction has a (projective) tensor factor whose degree is the | G : H | th root of the degree of D, does it follow that D is tensor induced from H? Among other results, examples are given to show that the answer can be negative when the index is 2. An affirmative answer is proved for normal subgroups of odd index, and also for arbitrary subgroups of odd prime index. As might be expected, the key lies in the study of F ( G ) / Z ( G ) as a symplectic module over a finite prime field; in particular, in exploring the connection between (ordinary) induction and form-induction of such modules.

Adhesional small bowel obstruction after colorectal surgery

Acute clinical indicators of complications in colorectal surgery can be readily attained but it is much harder to gather indicators of long-term outcomes such as small bowel obstruction (SBO). However, with improved in-hospital data collection and coding such information is becoming available. The aim of the present study was to examine our own hospital database for SBO rates post-colorectal surgery. The database was searched and then the relevant medical records were reviewed of all patients admitted to Flinders Medical Centre (Bedford Park, Australia) between July 1999 and November 2002 with a diagnosis of SBO following a colorectal procedure during this same time period. There were 21 patients that accounted for 28 readmissions from a total colorectal procedure group of 583 patients. The arbitrary subgroups were: 13/325 (4%) for colonic resections; 7/186 (3.7%) for rectal resections; 0/12 (0%) for laparotomies/other procedures; and 1/39 (2.6%) for stoma formation/reversals. The overall SBO rate requiring readmission was therefore 3.6%/pt in the 3 years. There was a large variation in the first readmission interval, 38% occurred within 3 months, 43% between 3 and 12 months, and 19% after 1 year. At the first readmission 38% of patients had operative treatment. The mean length of stay was 6.12 days for non-operative vs 21.62 days for operative treatment. The reported rate of SBO of 3.6% (at 3 years time interval) is in accordance with other studies. With 38% of patients being treated operatively at first admission there is good acceptance for conservative management in non-strangulated SBO. The prolonged hospital stay for patients needing surgery warrants further investigation.

Full duality for coactions of discrete groups

Using the strong relation between coactions of a discrete group $G$ on $C^*$-algebras and Fell bundles over $G$ we prove a new version of Mansfield's imprimitivity theorem for coactions of discrete groups. Our imprimitivity theorem works for the universally defined full crossed products and arbitrary subgroups of $G$ as opposed to the usual theory of [16], [11] which uses the spatially defined reduced crossed products and normal subgroups of $G$. Moreover, our theorem factors through the usual one by passing to appropriate quotients. As applications we show that a Fell bundle over a discrete group is amenable in the sense of Exel [7] if and only if the double dual action is amenable in the sense that the maximal and reduced crossed products coincide. We also give a new characterization of induced coactions in terms of their dual actions.

Some intersections and identifications in integral group rings

LetZG be the integral group ring of a groupG and I(G) its augmentation ideal. For a free groupF andR a normal subgroup ofF, the intersectionIn+1 (F) ∩In+1 (R) is determined for alln≥ 1. The subgroupsF ∩ (1+ZFI (R) I (F) I (S)) ANDF ∩ (1 + I (R)I3 (F)) of F are identified whenR and S are arbitrary subgroups ofF.

Supersymmetric E8(+8)/ SO(16) sigma-model coupled to N=1 supergravity in three dimensions

A three-dimensional simple N=1 supergravity theory with a supersymmetric sigma-model on the coset E 8(+8)/ SO(16) is constructed. Both bosons and fermions in the matter multiplets are in the spinorial 128 -representation of SO(16) with the same chirality. Due to their common chirality, this model cannot be obtained from the maximal N=16 supergravity. By introducing an independent vector multiplet, we can also gauge an arbitrary subgroup of SO(16) together with a Chern–Simons term. Similar N=1 supersymmetric σ-models coupled to supergravity are also constructed for the cosets F 4(−20)/ SO(9) and SO(8, n)/ SO(8)× SO( n).

Residual spectrum of odd orthogonal groups

The purpose of this paper is fourfold. First, we show that the Langlands' functorial lift of a generic cuspidal representation of SO 2n + 1 to GL 2n is of the form Ind σ1⊗…⊗σk, where σ i 's are (unitary) cuspidal representations of GL2ni such that L(s,σi,Λ2) has a pole at s=1, strengthening the recent result of Cogdell, Kim, Piatetski-Shapiro, and Shahidi. It uses the result of Barbasch and Moy on the spherical unitary dual,and the result of Ginzburg, Rallis, and Soudry that says that ifL(s,σ×t), where σ is a cuspidal representation of GLm and t is a generic cuspidal representation of SO2n+1, has a pole at s=1, then L(s,σ,Λ2) has a pole at s=1. Second, we give an application of the above result to the problem of determining the residual spectrum of SO2n+1 coming from maximal parabolic subgroups,with the assumption that the completed symmetric square L-function of GLn is holomorphic except possibly at s=0,1. Third, we describe the residual spectrum of SO5 and GSp4, defined over a number field. Even though Sp4 and SO5 are closely related (isogenous), their residual spectrums are quite different. Fourth, we give a conjectural description of the residual spectrum of SO2n+1 coming from arbitrary parabolic subgroups, assuming some standard global and local conjectures.

Relations between cusp forms on congruence and noncongruence groups

We construct a quadratic relation between cusp forms of weight two on four genus 1 subgroups of SL2 (Z). Two of the subgroups are congruence and two are noncongruence. We then generalize this to subgroups of F(N) of index 2. Let A be a finite-index subgroup of SL2(Z), and let XA be the associated Riemann surface (*/A. If K is a number field and C/K is a projective nonsingular curve, we say that C/K is a model for XA if there exists a morphism : C * JPl defined over K and a complex analytic isomorphism p: XA 0(QC) such that p(oo) is rational over K and XA P 0 (C) (1) 1 jjc * /SL2 (Z) i PI1 (C) commutes. (Here, the left hand arrow is the natural projection map.) It is known that for any A one can find a C/K as above, and, in fact, Belyi's theorem [B] tells us that the converse is true as well. In light of this, one may wish to study the usual arithmetic objects associated with congruence groups (i.e., Hecke operators, Fourier coefficients, and so forth) for arbitrary subgroups of finite index in SL2(Z). While it is true that on a general noncongruence subgroup, Hecke operators and Fourier coefficients are not related, there are nevertheless results concerning both. (See [Be] for Hecke operators on noncongruence subgroups and [S] for Fourier coefficients. From [Be], we see that Hecke operators on noncongruence subgroups give essentially no new information, but the Atkin-Swinnerton-Dyer congruences of [S] are highly nontrivial.) What has not been previously studied, however, is the relationship between cusp forms on related congruence and noncongruence groups. Since cusp forms on congruence groups are to some extent understood, any explicit information we obtain may be viewed as shedding light on the noncongruence forms. In particular, an actual polynomial relationship between forms would lead to an infinite number of relations between Fourier coefficients. One may view this as a Received by the editors November 16, 1998. 1991 Mathematics Subject Classification. Primary lIF1I; Secondary 1F30.

Supplementation in groups

In this paper, groups are investigated in which all subgroups, all normal subgroups, or all characteristic subgroups have a proper supplement. This supplement can be either an arbitrary subgroup, a normal or a characteristic subgroup, resulting in nine classes of groups. Properties of these classes are studied such as containment and closure properties, and characterizations for several of these classes are given.1991 Mathematics Subject Classification Primary 20E34, Secondary 20E15.

Wonderful blowups associated to group actions

A group action on a smooth variety provides it with the natural stratification by irreducible components of the fixed point sets of arbitrary subgroups. We show that the corresponding maximal wonderful blowup in the sense of MacPherson-Procesi has only abelian stabilizers. The result is inspired by the abelianization algorithm of Batyrev.

Variations of purity in abelian group theory

In a collection of classical papers, R. Nunke studied radicals R on the category of abelian groups constructed using extensions of the integers. In particular, the sequences in RExt AG were called R-pure. In this paper several types of purity related to R-purity are discussed. Projective and injective resolutions of an arbitrary group are constructed, providing interesting examples of dual classes of groups. One particular variation, that of R ∗ -purity, is shown to have many of the advantages of R-purity, while suffering from fewer of its drawbacks, e.g., every group has an R ∗ -injective hull and the class of R ∗ -projectives is closed under arbitrary subgroups. The most important example is R= p α , where α is an ordinal. When α is a limit, p α -purity and its generalizations are closely related to the completion of groups in the α-topology. Although these are the motivating examples, many of the results are stated in a substantially more general context.

On the concept of level for subgroups of SL2 over arithmetic rings

We define the concept of level for arbitrary subgroups Γ of finite index in the special linear group SL2(AS), whereAS is the ring ofS-integers of a global fieldk provided thatk is an algebraic number field, or card (S) ≥ 2. It is shown that this concept agrees with the usual notion of ‘Stufe’ for congruence subgroups. In the case SL2(O),O the ring of integers of an imaginary quadratic number field, this criterion for deciding whether or not an arbitrary subgroup of finite index is a congruence subgroup is used to determine the minimum of the indices of non-congruence subgroups.

Presentations for Subgroups of Monoids

The main result of this paper gives a presentation for an arbitrary subgroup of a monoid defined by a presentation. It is a modification of the well known Reidemeister–Schreier theorem for groups. Some consequences of this result are explored. It is proved that a regular monoid with finitely many left and right ideals is finitely presented if and only if all its maximal subgroups are finitely presented. An inverse monoid with finitely many left and right ideals is finitely presented as an inverse monoid if and only if it is finitely presented as a monoid. An example of a finitely presented monoid with a finitely generated but not finitely presented group of units is exhibited.

Morita Equivalent Blocks in Non-Normal Subgroups and p -Radical Blocks in Finite Groups

Let O be a complete discrete valuation ring with unique maximal ideal J(O), let K be its quotient field of characteristic 0, and let k be its residue field O/J(O) of prime characteristic p. We fix a finite group G, and we assume that K is big enough for G, that is, K contains all the ∣G∣-th roots of unity, where ∣G∣ is the order of G. In particular, K and k are both splitting fields for all subgroups of G. Suppose that H is an arbitrary subgroup of G. Consider blocks (block ideals) A and B of the group algebras RG and RH, respectively, where R∈{O, k}. We consider the following question: when are A and B Morita equivalent? Actually, we deal with ‘naturally Morita equivalent blocks A and B’, which means that A is isomorphic to a full matrix algebra of B, as studied by B. Külshammer. However, Külshammer assumes that H is normal in G, and we do not make this assumption, so we get generalisations of the results of Külshammer. Moreover, in the case H is normal in G, we get the same results as Külshammer; however, he uses the results of E. C. Dade, and we do not.