Part I. Kinetics of coarsening of the gamma-prime precipitate in nickel-silicon alloys. Part II. Rate of crystallization of an amorphous Fe-P-C alloy

<p>The coarsening kinetics of Ni₃ Si(��') precipitate in a binary Ni-Si alloy containing 6.5 wt. % silicon was studied by magnetic techniques and transmission electronmicroscopy. A calibration curve was established to determine the concentration of silicon in the matrix. The variation of the Si content of the Ni-rich matrix as a function of time follows Lifshitz and Wagner theory for diffusion controlled coarsening phenomena. The estimated values of equilibrium solubility of silicon in the matrix represent the true coherent equilibrium solubilities.</p> <p>The experimental particle-size distributions and average particle size were determined from dark field electron micrographs. The average particle size varies linearly with t^-1/3 as suggested by Lifshitz and Wagner. The experimental distributions of particle sizes differ slightly from the theoretical curve at the early stages of aging, but the agreement is satisfactory at the later stages. The values of diffusion coefficient of silicon, interfacial free energy and activation energy were calculated from the results of coarsening kinetics. The experimental value of effective diffusion coefficient is in satisfactory agreement with the value predicted by the application of irreversible the rmodynamics to the process of volume constrained growth of coherent precipitate during coarsening. The coherent ��' particles in Ni-Sialloy unlike those in Ni-Al and Ni-Ti seem to lose coherency at high temperature. A mechanism for the formation of semi-coherent precipitate is suggested.</p>

On the distribution of the signs of the conjugates of the cyclotomic units in the maximal real subfield of the qth cyclotomic field, q A prime

<p>Let F = ��(�� + ��^���1) be the maximal real subfield of the cyclotomic field ��(��) where �� is a primitive qth root of unity and q is an odd rational prime. The numbers u₁=-1, u_k=(��^k-��^-k)/(��-��^-1), k=2,���,p, p=(q-1)/2, are units in F and are called the cyclotomic units. In this thesis the sign distribution of the conjugates in F of the cyclotomic units is studied.</p> <p>Let G(F/��) denote the Galoi's group of F over ��, and let V denote the units in F. For each ���� G(F/��) and ����V define a mapping sgn_��: V���GF(2) by sgn_��(��) = 1 iff ��(��) �� 0 and sgn_��(��) = 0 iff ��(��) �� 0. Let {��₁, ... , ��_p} be a fixed ordering of G(F/��). The matrix M_q=(sgn_��j(v_i) ) , i, j = 1, ... , p is called the matrix of cyclotomic signatures. The rank of this matrix determines the sign distribution of the conjugates of the cyclotomic units. The matrix of cyclotomic signatures is associated with an ideal in the ring GF(2) [x] / (x^p+ 1) in such a way that the rank of the matrix equals the GF(2)-dimension of the ideal. It is shown that if p = (q-1)/ 2 is a prime and if 2 is a primitive root mod p, then M_q is non-singular. Also let p be arbitrary, let ��� be a primitive root mod q and let L = {i | 0 ��� i ��� p-1, the least positive residue of defined by ���ⁱ mod q is greater than p}. Let H_q(x) �� GF(2)[x] be defined by H_q(x) = g. c. d. ((�� xⁱ/I �� L) (x+1) + 1, x^p + 1). It is shown that the rank of M_q equals the difference p - degree H_q(x).</p> <p>Further results are obtained by using the reciprocity theorem of class field theory. The reciprocity maps for a certain abelian extension of F and for the infinite primes in F are associated with the signs of conjugates. The product formula for the reciprocity maps is used to associate the signs of conjugates with the reciprocity maps at the primes which lie above (2). The case when (2) is a prime in F is studied in detail. Let T denote the group of totally positive units in F. Let U be the group generated by the cyclotomic units. Assume that (2) is a prime in F and that p is odd. Let F₍₂₎ denote the completion of F at (2) and let V₍₂₎ denote the units in F₍₂₎. The following statements are shown to be equivalent. 1) The matrix of cyclotomic signatures is non-singular. 2) U���T = U². 3) U���F²₍₂₎ = U². 4) V₍₂₎/ V₍₂₎² = ��<i>v</i>₁ V₍₂₎²�� ���������<i>v</i>_p V₍₂₎²�� �� ��3V₍₂₎²��.</p> <p>The rank of M_q was computed for 5���q���929 and the results appear in tables. On the basis of these results and additional calculations the following conjecture is made: If q and p = (q -1)/ 2 are both primes, then M_q is non-singular. </p>

On the Mumford-Tate conjecture for Abelian varieties with reduction conditions

<p>In this thesis we study Galois representations corresponding to abelian varieties with certain reduction conditions. We show that these conditions force the image of the representations to be "big," so that the Mumford-Tate conjecture (:= MT) holds. We also prove that the set of abelian varieties satisfying these conditions is dense in a corresponding moduli space. </p> <p>The main results of the thesis are the following two theorems. </p> <p>Theorem A: Let A be an absolutely simple abelian variety, End�� (A) = k : imaginary quadratic field, g = dim(A). Assume either dim(A) ��� 4, or A has bad reduction at some prime ��, with the dimension of the toric part of the reduction equal to 2r, and gcd(r,g) = 1, and (r,g) ��� (15,56) or (m -1, m(m+1)/2). Then MT holds. </p> <p>Theorem B: Let M be the moduli space of abelian varieties with fixed polarization, level structure and a k-action. It is defined over a number field F. The subset of M(Q) corresponding to absolutely simple abelian varieties with a prescribed stable reduction at a large enough prime �� of F is dense in M(C) in the complex topology. In particular, the set of simple abelian varieties having bad reductions with fixed dimension of the toric parts is dense. </p> <p>Besides this we also established the following results: </p> <p> (1) MT holds for some other classes of abelian varieties with similar reduction conditions. For example, if A is an abelian variety with End�� (A) = Q and the dimension of the toric part of its reduction is prime to dim( A), then MT holds. </p> <p> (2) MT holds for Ribet-type abelian varieties. </p> <p> (3) The Hodge and the Tate conjectures are equivalent for abelian 4-folds. </p> <p> (4) MT holds for abelian 4-folds of type II, III, IV (Theorem 5.0(2)) and some 4-folds of type I. </p> <p> (5) For some abelian varieties either MT or the Hodge conjecture holds. </p>

Arithmetic of cyclotomic fields and Fermat-type equations

<p>Let l be any odd prime, and �� a primitive l-th root of unity. Let C_l be the l-Sylow subgroup of the ideal class group of Q(��). The Teichm��ller character w : Z_l ��� Z^*_l is given by w(x) = x (mod l), where w(x) is a p-1-st root of unity, and x ��� Z_l. Under the action of this character, C_l decomposes as a direct sum of C^((i))_l, where C^((i))_l is the eigenspace corresponding to w^i. Let the order of C^((3))_l be l^h_3). The main result of this thesis is the following: For every n ��� max( 1, h_3 ), the equation x^(ln) + y^(ln) + z^(ln) = 0 has no integral solutions (x,y,z) with l ��� xyz. The same result is also proven with n ��� max(1,h_5), under the assumption that C_l^((5)) is a cyclic group of order l^h_5. Applications of the methods used to prove the above results to the second case of Fermat's last theorem and to a Fermat-like equation in four variables are given.</p> <p>The proof uses a series of ideas of H.S. Vandiver ([Vl],[V2]) along with a theorem of M. Kurihara [Ku] and some consequences of the proof of lwasawa's main conjecture for cyclotomic fields by B. Mazur and A. Wiles [MW]. In [V1] Vandiver claimed that the first case of Fermat's Last Theorem held for l if l did not divide the class number h^+ of the maximal real subfield of Q(e^(2��i/i)). The crucial gap in Vandiver's attempted proof that has been known to experts is explained, and complete proofs of all the results used from his papers are given.</p>

Earthquakes of the Nepal Himalaya : towards a physical model of the seismic cycle

Home to hundreds of millions of souls and land of excessiveness, the Himalaya is also the locus of a unique seismicity whose scope and peculiarities still remain to this day somewhat mysterious. Having claimed the lives of kings, or turned ancient timeworn cities into heaps of rubbles and ruins, earthquakes eerily inhabit Nepalese folk tales with the fatalistic message that nothing lasts forever. From a scientific point of view as much as from a human perspective, solving the mysteries of Himalayan seismicity thus represents a challenge of prime importance. Documenting geodetic strain across the Nepal Himalaya with various GPS and leveling data, we show that unlike other subduction zones that exhibit a heterogeneous and patchy coupling pattern along strike, the last hundred kilometers of the Main Himalayan Thrust fault, or MHT, appear to be uniformly locked, devoid of any of the ���creeping barriers��� that traditionally ward off the propagation of large events. The approximately 20 mm/yr of reckoned convergence across the Himalaya matching previously established estimates of the secular deformation at the front of the arc, the slip accumulated at depth has to somehow elastically propagate all the way to the surface at some point. And yet, neither large events from the past nor currently recorded microseismicity nearly compensate for the massive moment deficit that quietly builds up under the giant mountains. Along with this large unbalanced moment deficit, the uncommonly homogeneous coupling pattern on the MHT raises the question of whether or not the locked portion of the MHT can rupture all at once in a giant earthquake. Univocally answering this question appears contingent on the still elusive estimate of the magnitude of the largest possible earthquake in the Himalaya, and requires tight constraints on local fault properties. What makes the Himalaya enigmatic also makes it the potential source of an incredible wealth of information, and we exploit some of the oddities of Himalayan seismicity in an effort to improve the understanding of earthquake physics and cipher out the properties of the MHT. Thanks to the Himalaya, the Indo-Gangetic plain is deluged each year under a tremendous amount of water during the annual summer monsoon that collects and bears down on the Indian plate enough to pull it away from the Eurasian plate slightly, temporarily relieving a small portion of the stress mounting on the MHT. As the rainwater evaporates in the dry winter season, the plate rebounds and tension is increased back on the fault. Interestingly, the mild waggle of stress induced by the monsoon rains is about the same size as that from solid-Earth tides which gently tug at the planets solid layers, but whereas changes in earthquake frequency correspond with the annually occurring monsoon, there is no such correlation with Earth tides, which oscillate back-and-forth twice a day. We therefore investigate the general response of the creeping and seismogenic parts of MHT to periodic stresses in order to link these observations to physical parameters. First, the response of the creeping part of the MHT is analyzed with a simple spring-and-slider system bearing rate-strengthening rheology, and we show that at the transition with the locked zone, where the friction becomes near velocity neutral, the response of the slip rate may be amplified at some periods, which values are analytically related to the physical parameters of the problem. Such predictions therefore hold the potential of constraining fault properties on the MHT, but still await observational counterparts to be applied, as nothing indicates that the variations of seismicity rate on the locked part of the MHT are the direct expressions of variations of the slip rate on its creeping part, and no variations of the slip rate have been singled out from the GPS measurements to this day. When shifting to the locked seismogenic part of the MHT, spring-and-slider models with rate-weakening rheology are insufficient to explain the contrasted responses of the seismicity to the periodic loads that tides and monsoon both place on the MHT. Instead, we resort to numerical simulations using the Boundary Integral CYCLes of Earthquakes algorithm and examine the response of a 2D finite fault embedded with a rate-weakening patch to harmonic stress perturbations of various periods. We show that such simulations are able to reproduce results consistent with a gradual amplification of sensitivity as the perturbing period get larger, up to a critical period corresponding to the characteristic time of evolution of the seismicity in response to a step-like perturbation of stress. This increase of sensitivity was not reproduced by simple 1D-spring-slider systems, probably because of the complexity of the nucleation process, reproduced only by 2D-fault models. When the nucleation zone is close to its critical unstable size, its growth becomes highly sensitive to any external perturbations and the timings of produced events may therefore find themselves highly affected. A fully analytical framework has yet to be developed and further work is needed to fully describe the behavior of the fault in terms of physical parameters, which will likely provide the keys to deduce constitutive properties of the MHT from seismological observations.

An arithmetical theorem for partially ordered sets

<p>The simplest multiplicative systems in which arithmetical ideas can be defined are semigroups. For such systems irreducible (prime) elements can be introduced and conditions under which the fundamental theorem of arithmetic holds have been investigated (Clifford (3)). After identifying associates, the elements of the semigroup form a partially ordered set with respect to the ordinary division relation. This suggests the possibility of an analogous arithmetical result for abstract partially ordered sets. Although nothing corresponding to product exists in a partially ordered set, there is a notion similar to g.c.d. This is the meet operation, defined as greatest lower bound. Thus irreducible elements, namely those elements not expressible as meets of proper divisors can be introduced. The assumption of the ascending chain condition then implies that each element is representable as a reduced meet of irreducibles. The central problem of this thesis is to determine conditions on the structure of the partially ordered set in order that each element have a unique such representation.</p> <p>Part I contains preliminary results and introduces the principal tools of the investigation. In the second part, basic properties of the lattice of ideals and the connection between its structure and the irreducible decompositions of elements are developed. The proofs of these results are identical with the corresponding ones for the lattice case (Dilworth (2)). The last part contains those results whose proofs are peculiar to partially ordered sets and also contains the proof of the main theorem.</p>

Phospholipids and a protein-conducting channel regulate cotranslational protein targeting

<p>Signal recognition particle (SRP) and signal recognition particle receptor (SR) are evolutionarily conserved GTPases that deliver secretory and membrane proteins to the protein-conducting channel Sec61 complex in the lipid bilayer of the endoplasmic reticulum in eukaryotes or the SecYEG complex in the inner membrane of bacteria. Unlike the canonical Ras-type GTPases, SRP and SR are activated via nucleotide-dependent heterodimerization. Upon formation of the SR���SRP targeting complex, SRP and SR undergo a series of discrete conformational changes that culminate in their reciprocal activation and hydrolysis of GTP. How the SR���SRP GTPase cycle is regulated and coupled to the delivery of the cargo protein to the protein-conducting channel at the target membrane is not well-understood. Here we examine the role of the lipid bilayer and SecYEG in regulation of the SRP-mediated protein targeting pathway and show that they serve as important biological cues that spatially control the targeting reaction. </p> <p>In the first chapter, we show that anionic phospholipids of the inner membrane activate the bacterial SR, FtsY, and favor the late conformational states of the targeting complex conducive to efficient unloading of the cargo. The results of our studies suggest that the lipid bilayer acts as a spatial cue that weakens the interaction of the cargo protein with SRP and primes the complex for unloading its cargo onto SecYEG. </p> <p>In the second chapter, we focus on the effect of SecYEG on the conformational states and activity of the targeting complex. While phospholipids prime the complex for unloading its cargo, they are insufficient to trigger hydrolysis of GTP and the release of the cargo from the complex. SecYEG modulates the conformation of the targeting complex and triggers the GTP hydrolysis from the complex, thus driving the targeting reaction to completion. The results of this study suggest that SecYEG is not a passive recipient of the cargo protein; rather, it actively releases the cargo from the targeting complex. Together, anionic phospholipids and SecYEG serve distinct yet complementary roles. They spatially control the targeting reaction in a sequential manner, ensuring efficient delivery and unloading of the cargo protein. </p> <p>In the third chapter, we reconstitute the transfer reaction in vitro and visualize it in real time. We show that the ribosome-nascent chain complex is transferred to SecYEG via a stepwise mechanism with gradual dissolution and formation of the contacts with SRP and SecYEG, respectively, explaining how the cargo is kept tethered to the membrane during the transfer and how its loss to the cytosol is avoided. </p> <p>In the fourth chapter, we examine interaction of SecYEG with secretory and membrane proteins and attempt to address the role of a novel insertase YidC in this interaction. We show that detergent-solubilized SecYEG is capable of discriminating between the nascent chains of various lengths and engages a signal sequence in a well-defined conformation in the absence of accessory factors. Further, YidC alters the conformation of the signal peptide bound to SecYEG. The results described in this chapter show that YidC affects the SecYEG-nascent chain interaction at early stages of translocation/insertion and suggest a YidC-facilitated mechanism for lateral exit of transmembrane domains from SecYEG into the lipid bilayer. </p>

Structure-Guided Design and Biophysical Characterization of Novel Anti-HIV Reagents

<p>Despite over 30 years of effort, an HIV-1 vaccine that elicits protective antibodies still does not exist. Recent clinical studies have identified that during natural infection about 20% of the population is capable of mounting a potent and protective antibody response. Closer inspection of these individuals reveal that a subset of these antibodies, recently termed potent VRC01-like (PVL), derive exclusively from a single human germline heavy chain gene. Induced clonal expansion of the B cell encoding this gene is the first step through which PVL antibodies may be elicited. Unfortunately, naturally occurring HIV gp120s fail to bind to this germline, and as a result cannot be used as the initial prime for a vaccine regimen. We have determined the crystal structure of an important germline antibody that is a promising target for vaccine design efforts, and have set out to engineer a more likely candidate using computationally-guided rational design. </p> <p>In addition to prevention efforts on the side of vaccine design, recently characterized broadly neutralizing anti-HIV antibodies have excellent potential for use in gene therapy and passive immunotherapy. The separation distance between functional Fabs on an antibody is important due to the sparse distribution of envelop spikes on HIV compared to other viruses. We set out to build and characterize novel antibody architectures by incorporating structured linkers into the hinge region of an anti-HIV antibody b12. The goal was to observe whether these linkers increased the arm-span of the IgG dimer. When incorporated, flexible Gly4Ser repeats did not result in detectable extensions of the IgG antigen binding domains, by contrast to linkers including more rigid domains such as ��2-microglobulin, Zn-��2-glycoprotein, and tetratricopeptide repeats (TPRs). This study adds an additional set of linkers with varying lengths and rigidities to the available linker repertoire, which may be useful for the modification and construction of antibodies and other fusion proteins. </p>

Development of Metalloenzyme Dioxygen Reduction Cathodes

<p>The prime thrust of this dissertation is to advance the development of fuel cell dioxygen reduction cathodes that employ some variant of multicopper oxidase enzymes as the catalyst. The low earth-abundance of platinum metal and its correspondingly high market cost has prompted a general search amongst chemists and materials scientists for reasonable alternatives to this metal for facilitating catalytic dioxygen reduction chemistry. The multicopper oxidases (MCOs), which constitute a class of enzyme that naturally catalyze the reaction O₂ + 4H⁺ + 4e^- ��� 2H₂O, provide a promising set of biochemical contenders for fuel cell cathode catalysts. In MCOs, a substrate reduces a copper atom at the type 1 site, where charge is then transferred to a trinuclear copper cluster consisting of a mononuclear type 2 or ���normal copper��� site and a binuclear type 3 copper site. Following the reduction of all four copper atoms in the enzyme, dioxygen is then reduced to water in two two-electron steps, upon binding to the trinuclear copper cluster. We identified an MCO, a laccase from the hyperthermophilic bacterium Thermus thermophilus strain HB27, as a promising candidate for cathodic fuel cell catalysis. This protein demonstrates resilience at high temperatures, exhibiting no denaturing transition at temperatures high as 95��C, conditions relevant to typical polymer electrolyte fuel cell operation.</p> <p>In Chapter I of this thesis, we discuss initial efforts to physically characterize the enzyme when operating as a heterogeneous cathode catalyst. Following this, in Chapter II we then outline the development of a model capable of describing the observed electrochemical behavior of this enzyme when operating on porous carbon electrodes. Developing a rigorous mathematical framework with which to describe this system had the potential to improve our understanding of MCO electrokinetics, while also providing a level of predictive power that might guide any future efforts to fabricate MCO cathodes with optimized electrochemical performance. In Chapter III we detail efforts to reduce electrode overpotentials through site-directed mutagenesis of the inner and outer-sphere ligands of the Cu sites in laccase, using electrochemical methods and electronic spectroscopy to try and understand the resultant behavior of our mutant constructs. Finally, in Chapter IV, we examine future work concerning the fabrication of enhanced MCO cathodes, exploring the possibility of new cathode materials and advanced enzyme deposition techniques.</p>

Structural Characterizations of the Dimeric Anti-HIV Antibody 2G12 and the HIV-2 Envelope Glycoprotein

More than thirty years after the discovery that Human Immunodeficiency Virus (HIV) was the causative agent of Acquired Immunodeficiency Syndrome (AIDS), the disease remains pandemic as long as no effective universal vaccine is found. Over 34 million individuals in the world are infected with the virus, and the vast majority of them have no access to the antiretroviral therapies that have largely reduced HIV to a chronic disease in the developed world. The first chapter of this thesis introduces the history of the virus. The key to the infectious mechanism of the virus lies in its envelope glycoprotein (Env), a trimeric spike on the viral surface that utilizes host T cell receptors for entry. Though HIV-1 Env is immunogenic, most infected patients do not mount an effective neutralizing antibody response against it. Broadly-neutralizing anti-Env antibodies (bNAbs) present in the serum of a minority of infected individuals are usually sufficient to prevent the progression to full blown AIDS. Thus, the molecular details of these bNAbs as well as the antibody-antigen interface are of prime interest for structural studies, as insight gained would contribute to the design of a more effective immunogen and potential vaccine candidate. The second chapter of this thesis describes the low-resolution crystal structure of one such antibody, 2G12 dimer, which targets a high mannose epitope on the surface of Env. Patients infected with HIV-2, a related virus with ~35% sequence identity in the Env region, can generally mount a robust antibody response sufficient for viral control for reasons still unknown. The final two chapters of this thesis focus on the first reported structural studies of HIV-2 Env, the molecular details of which may inform HIV-1 therapy and immunogen design.

Special Frobenius Traces in Galois Representations

<p>This thesis studies Frobenius traces in Galois representations from two different directions. In the first problem we explore how often they vanish in Artin-type representations. We give an upper bound for the density of the set of vanishing Frobenius traces in terms of the multiplicities of the irreducible components of the adjoint representation. Towards that, we construct an infinite family of representations of finite groups with an irreducible adjoint action.</p> <p>In the second problem we partially extend for Hilbert modular forms a result of Coleman and Edixhoven that the Hecke eigenvalues a_p of classical elliptical modular newforms f of weight 2 are never extremal, i.e., a_p is strictly less than 2[square root]p. The generalization currently applies only to prime ideals p of degree one, though we expect it to hold for p of any odd degree. However, an even degree prime can be extremal for f. We prove our result in each of the following instances: when one can move to a Shimura curve defined by a quaternion algebra, when f is a CM form, when the crystalline Frobenius is semi-simple, and when the strong Tate conjecture holds for a product of two Hilbert modular surfaces (or quaternionic Shimura surfaces) over a finite field.</p>

Studies on the role of histones in the structure and function of chromatin

<p>Studies on the dissociation of histones from chromatin by increasing concentrations of sodium deoxycholate (DOC) have shown that histrone II is removed at lowest concentrations of DOC, while slightly higher concentrations remove histones III and IV. Still higher concentrations remove histone I.</p> <p>The complete separation of chromatin and ¹⁴C-DOC by sucrose sedimentation indicated that the binding of DOC to chromatin is readily and completely reversible.</p> <p>The dissociation of histones from chromatin by increasing concentrations of related cholanic acids and some of their conjugated derivatives were studied. The results suggested that the driving force for the interaction between the cholanic acid anion and histones is the lowering of the activity coefficient of the cholanic acid anion which occurs when it is partially removed from solution by interaction with hydrophobic regions of the positively charged histones.</p> <p>The role of histones in the structure of chromatin has been studied by comparing the effects of selective removal of histones from chromatin by increasing concentrations of DOC with those caused by NaCl (removes histone I at lowest concentrations, while higher concentrations remove histones II, III, and IV). Properties studied included thermal denaturation, sedimentation velocity, flow dichroism, relaxation times of molecules oriented in a flow field, and the irreversible disruption of a 130 S, cross-linked component of sheared chromatin. The data indicated that none of the structural or chemical parameters with which these properties are correlated show a dependence on the presence of one particular histone fraction. </p> <p>The template activity (ability to prime a 0.2 <u>M</u> KC1 DNA-dependent RNA synthesis system catalyzed by <u>E</u>. <u>coli</u> RNA polymerase) increases from that of native chromatin (approximately 25 per cent of that pure DNA) to that of pure DNA in a fashion which shows a nearly linear relationship to the amount of histone coverage of the template. The precipitability of partially dehistonized chromatin samples in 0.15 <u>M</u> NaCl shows a large dependence on the presence of histone I. </p>

On an embedding property of generalized Carter subgroups

<p>If <i>E</i> and <i>F</i> are saturated formations, we say that <i>E</i> is strongly contained in <i>F</i> if for any solvable group G with <i>E</i>-subgroup, E, and <i>F</i>-subgroup, F, some conjugate of E is contained in F. In this paper, we investigate the problem of finding the formations which strongly contain a fixed saturated formation <i>E</i>.</p> <p>Our main results are restricted to formations, <i>E</i>, such that <i>E</i> = {G|G/F(G) ��<i>T</i>}, where <i>T</i> is a non-empty formation of solvable groups, and F(G) is the Fitting subgroup of G. If <i>T</i> consists only of the identity, then <i>E</i>=<i>N</i>, the class of nilpotent groups, and for any solvable group, G, the <i>N</i>-subgroups of G are the Carter subgroups of G.</p> <p>We give a characterization of strong containment which depends only on the formations <i>E</i>, and <i>F</i>. From this characterization, we prove:</p> <p>If <i>T</i> is a non-empty formation of solvable groups, <i>E</i> = {G|G/F(G) ��<i>T</i>}, and <i>E</i> is strongly contained in <i>F</i>, then </p> <p>(1) there is a formation <i>V</i> such that <i>F</i> = {G|G/F(G) ��<i>V</i>}.</p> <p>(2) If for each prime p, we assume that <i>T</i> does not contain the class, <i>S</i>_p���, of all solvable p���-groups, then either <i>E</i> = <i>F</i>, or <i>F</i> contains all solvable groups.</p> <p>This solves the problem for the Carter subgroups. </p> <p>We prove the following result to show that the hypothesis of (2) is not redundant:</p> <p>If <i>R</i> = {G|G/F(G) ��<i>S</i>_r���}, then there are infinitely many formations which strongly contain <i>R</i>.</p>

Class two p groups as fixed point free automorphism groups

<p>Suppose that AG is a solvable group with normal subgroup G where (|A|, |G|) = 1. Assume that A is a class two odd p group all of whose irreducible representations are isomorphic to subgroups of extra special p groups. If p^c ��� r^d + 1 for any c = 1, 2 and any prime r where r^2d+1 divides |G| and if C_G(A) = 1 then the Fitting length of G is bounded by the power of p dividing |A|.</p> <p>The theorem is proved by applying a fixed point theorem to a reduction of the Fitting series of G. The fixed point theorem is proved by reducing a minimal counter example. IF R is an extra spec r subgroup of G fixed by A₁, a subgroup of A, where A₁ centralizes D(R), then all irreducible characters of A₁R which are nontrivial on Z(R) are computed. All nonlinear characters of a class two p group are computed. </p>