Ordinary Character Research Articles

A Note on Lifting Brauer Characters

A Brauer character of a finite group may be lifted to an ordinary character if it lies in a block whose defect groups are contained in a normal $p$-solvable subgroup.

Blocks with a cyclic defect group

This thesis is concerned with the block theory of a finite group over an algebraically closed field of prime characteristic. We study the indecomposable modules in such a block Ḇ.with a cyclic defect group, and then the results that we get are applied to the principal p-blocks of the symmetric group Sp' the alternating group ~ and the five Mathieu groups, Ap being an odd prime. There are three parts to this thesis, each one dealing with the following topics. Part A is concerned with finding the full submodule lattice of each one of the projective indecomposable modules in Ḇ. These lattices turn out to have simple diamond shapes, and depend upon parameters r(i),s(i). In part B we generalise the methods used in the previous part to get a good description of a full set of non-projective indecomposables in Ḇ.(though we do not manage to get their full submodule lattices). Finally in part C we show how to use ordinary character theory, and in particular the Brauer tree, to work out these positive integers r(i),s(i) and then we apply such methods to work out the lattices of those examples listed above.

A note on lifting Brauer characters

A Brauer character of a finite group may be lifted to an ordinary character if it lies in a block whose defect groups are contained in a normal p p -solvable subgroup.

Modular permutation representations

A modular theory for permutation representations and their centralizer rings is presented, analogous in several respects to the classical work of Brauer on group algebras. Some principal ingredients of the theory are characters of indecomposable components of the permutation module over a p-adic ring, modular characters of the centralizer ring, and the action of normalizers of p-subgroups P on the fixed points of P. A detailed summary appears in [15]. A main consequence of the theory is simplification of the problem of computing the ordinary character table of a given centralizer ring. Also, some previously unsuspected properties of permutation characters emerge. Finally, the theory provides new insight into the relation of Brauer’s theory of blocks to Green’s work on indecomposable modules.

Modular Permutation Representations

A modular theory for permutation representations and their centralizer rings is presented, analogous in several respects to the classical work of Brauer on group algebras. Some principal ingredients of the theory are characters of indecomposable components of the permutation module over a p-adic ring, modular characters of the centralizer ring, and the action of normalizers of p-subgroups P on the fixed points of P. A detailed summary appears in [15]. A main consequence of the theory is simplification of the problem of computing the ordinary character table of a given centralizer ring. Also, some previously unsuspected properties of permutation characters emerge. Finally, the theory provides new insight into the relation of Brauer’s theory of blocks to Green’s work on indecomposable modules.

Chorus and Character in “Samson Agonistes”

ABSTRACT The tragic action of Samson Agonistes develops through a contrast between the language of Samson and that of the Chorus. An essential clue to Milton's art in this drama lies in his effect of deliberately playing down the choruses in the middle, refusing to give the chorus very good lines, except toward the beginning and again toward the end. After each of the central episodes, Milton has given the chorus words that are for the most part dry and commonplace, often surprisingly flat. It would appear that Milton's use of rhyme and partial rhyme in the choruses has frequently the effect of lowering the tone of high seriousness and making impossible any high dignity in the choric speech. The basic effect is to stress, by contrast, the grandeur of Samson, the greatness of his mind and spirit. The weak and ordinary character of Manoa, the complex and enigmatic character of Dalila, and the braggart character of Harapha are all related to this basic contrast between Samson and the chorus.

Quadratic and higher reciprocity of modular polynomials

1. We first compare the λ-ic characters of two polynomials with respect to each other and the prime p, the polynomials being irreducible with respect to p. This is done by relating each to the resultant of the polynomials. We then show in § 6 that the ordinary quadratic character of one prime with respect to another is also the resultant of two polynomials.

William Mackie, M.A., M.D., LL.D., physician and geologist. [Obituary.

William Mackie was born on 28th April 1856 at Loanends of Durno in the parish of Chapel of Gariock, Aberdeenshire. He received his early education first at the Parish School and afterwards at the Old Aberdeen Grammar School. After undergoing a period of training as a pupil teacher at Daviot under the late Dr Brebner, he proceeded to Aberdeen University where he graduated in Arts with honours in Natural Science. For some years afterwards he held posts as a teacher of science in schools in Ayr and Perth. While teaching in Perth he found time to attend classes in chemistry in the University College, Dundee, and also to carry out research work there in collaboration with Professor Thomas Carnelley. Having decided to enter the medical profession Mackie returned to Aberdeen University and graduated M.B., Ch.B. in 1888. Following a period of postgraduate study in Aberdeen and Edinburgh, during which he made a special study of diseases of the eye, Dr Mackie settled in Elgin. There he carried on a general medical practice for over thirty-five years, and for eighteen years he held also the posts of Medical Officer of Health for Elgin and Rothes. In 1924 Dr Mackie retired from active practice and thereafter resided in Glasgow where he died on 15th July 1932. Endowed with intellectual ability of no ordinary character, and possessing marked versatility, Mackie was withal an indefatigable worker, and he found time, despite the ties of a successful medical practice, to carry out an extraordinarily large

The Vows of Baldwin

Although one of the most interesting of Middle English romances, the Avowing of Arthur has been singularly neglected. The story is told with the gratifying freshness which marks Gawain and the Green Knight and the other poems of the Northern school. The incidents of which the story is composed are fitted into the general framework with rare skill. Even more striking is the vividness of the characterization. The ordinary romance character is a dummy upon which are hung splendid clothes tagged with catalogues of all the virtues. Dealing with these personages is often like handling the bits of cardboard stamped “sugar,” “tea,” “potatoes,” with which students in commercial colleges play. But in this romance there is sharp distinction between Arthur, genial, brave, a practical joker, and Baldwin, a man of few words, cynical without being bitter, nonchalant, a man of deeds; between Kay, impulsive, always getting into scrapes, inclined to jeer at others, a great boy with a boy's love of adventure, and Gawain, the courteous knight, equally ready to aid beauty in distress and to assist a comrade in time of need. The story is crowded with incidents, and the verse is vigorous and effective.

The Assumed Radio-activity of Ordinary Materials

WITH reference to Mr. Strutt's recent article and Prof. J. J. Thomson's letter on this subject, may I venture modestly to urge that it may be well to consider whether the condition set up in air to which attention is directed be not the outcome of the occurrence of a minute amount of chemical change of an ordinary character—whether it be not a sort of Russell effect on an infinitely minute scale, detected by an infinitely delicate test? That oxidative change is in continual progress, I imagine, is the belief of everyone who has paid the slightest attention to the subject; and that leaf surfaces—if not waterfalls—are the certain seat of such changes may be regarded as unquestionable. Those of us who require something more than an attitude of papal infallibility in proof of a scientific proposition would like to see the old love honourably retired before the new one is accepted in society.

ON TUBAGE AS ACCESSORY TREATMENT IN CERTAIN CASES OF SEPTAL DEFORMITY.Read in the Section on Laryngology and Otology at the Forty-fifth Annual Meeting of the American Medical Association, held at San Francisco. June 5-8, 1894.

Deviations of the nasal septum are of such frequent occurrence among all civilized races that we need not wonder at the number and variety of methods practiced for their removal. Each method of operation is often eulogized by its advocates as superior to all others, and histories of cases are cited as direct evidence of the truth of the statement. Still, while granting all that can be said in favor of the many ways of treatment, every operator will admit, I think, that cases sometimes occur in which mere operative measures are insufficient, unless supported by accessory treatment of a more than ordinary character. Judging from personal experience, I think this is particularly true of that class of cases, characterized by extensive adhesions between the septum and middle and inferior turbinated bones. I do not refer to cases where synechiæ or cicatricial bands, due to inflammatory action, are stretched across

V.—Two New French Meteorites

Two Meteorites from France have recently come into my possession from a private source, one of which, though not of a recent fall (1836), has neither been described nor chemically examined, and which I now briefly bring to notice, it being of a most rare type. The other is of a much more recent fall (1875), and is of a much more ordinary character, being similar to many other meteorites of the same class; this stone also appears not to have been described, nor has any notice of it been published. It is somewhat a remarkable coincidence that these two stones should have fallen in the same Department, although at such widely different dates, and from the data and notes I have received from careful inquiries, I now bring forward the following details of these two aerolites.

Interglacial Deposits of West Cumberland and North Lancashire

Seldom do we find the whole three members present in one section. Sometimes the Lower Boulder-clay alone is found ; at others this is overlain by sand or gravel or clay, or by some or all of these rocks. In other cases we find the series complete. They occur almost continuously, in more or less completeness, from the sea-shore to an altitude of 500 feet ; and from that level they appear in patches up to 1000 feet above the sea. The two Boulder-clays have the ordinary character, and are very much alike, except that the lower is tougher than the upper and contains larger boulders. The character of the included stones is the same in both clays, as well as in the middle sands and gravel.Some most remarkable facts are presented by the distribution of these boulders. Associated with these glacial beds, and occurring at various places, sometimes inland, sometimes along the sea-coast, between high- and low-water marks, there are a number of deposits of vegetable matter, which hitherto have been almost entirely neglected by geologists. When occurring on the sea-shore, these deposits usually pass by the name of “submerged forests.” I am, however, inclined to doubt the accuracy of this appellation. After a long and careful investigation into the nature of these deposits, I have come to the conclusion that they are not forests at all, nor the sites of forests,as will appear further on. 2. Observed Facts Lindal deposit . —The first of these deposits to which I shall

I. On the lunar variations of magnetic declination at Bombay

This paper is in continuation of that “On the Solar Variations of Magnetic Declination at Bombay,” published in the Philosophical Transactions for 1869; but the discussion is confined to the observations of the years 1861 to 1863, which alone have as yet been reduced. The point of principal interest brought out in the discussion is that whilst the mean lunar-diurnal variation is of the ordinary character, having as its principal feature a double oscillation in the lunar day, its range is very small as compared with the several ranges of the lunar-diurnal variations when the sun and moon have several specific varieties of relative position; and moreover, although in those latter variations the double oscillation is generally preserved as a main feature, correspondence of phase in the representative curves is as generally absent; and in some cases the curves are, whilst systematic, altogether different in character from the mean lunar-diurnal variation curve. The semiannual inequality in the lunar-diurnal variation, whilst it is as definitely systematic, has twice the range of the mean lunar, diurnal variation; and it is also subject to remarkable modifications which accompany changes of phase of the moon.

I. Lord Coningsby's Account of the State of Political Parties during the Reign of Queen Anne

Among the Lansdowne Manuscripts in the British Museum there is a short Memoir of the State of Political Parties in England from the last year of the reign of William the Third to the death of Queen Anne, written by the Earl of Coningsby (at least so afterwards created), who was throughout engaged in the ministerial divisions of that period, and divisions they were of no ordinary character.

An Account of some Experiments with the Steam Jet as a Ventilator for Mines

A few months ago, the writer’s attention was drawn to certain alterations required in the pumps employed in draining a small colliery in this neighbourhood. The shaft is about 120 yards deep, and the nature of the alterations was such as to put a stop to the working of the pumps for some time; during this stoppage the water accumulated in the workings of the mine, and ascended some distance up the shaft, thus completely stopping the system of ventilation previously employed. The ventilation of the pit had no sooner ceased, than the accumulation of carbonic acid gas (or choke damp) rendered it impossible for the workmen to proceed. Several attempts of an ordinary character were made to displace the gas but they did not succeed, and there appeared a probability of having to incur some expense before proper ventilation of the pit could be accomplished. Matters remained in this state for several days, and further attempts were made to get rid of the gas, but still without the desired effect. The completion of the pumps now became urgent, and therefore, the removal of the noxious gas indispensable. After careful examination of what had been done, and finding it all practically correct, and such as had before been found sufficient for the purpose in most ordinary cases of a similar nature, the writer determined upon trying the effect of a jet of steam, the application of which happened to be exceedingly favourable; and the result was the most complete success; ...

Art. XVIII.—Arabic Inscription from China

An Inscription on fine Chinese paper was, some months since, presented to the Society by one of its Members, John Romer, Esq., to whom it had been given by the Captain of a vessel returning from the East, who did not remember how or where he had obtained it. It is four feet eight inches in length, by two feet four inches in width; and a reduced copy of it is herewith given. After it had been suspended a short time in the Society's Meeting Room, a letter was received by the Director from John Shakespear, Esq., with a transcript of this singular document in the ordinary character, and a translation. It is not easy to account for the origin of this inscription. It was certainly done in China, as appears from the stamps attached to it, which were made before the inscription was written, unless room was purposely left for placing them. The characters are drawn with freedom, as though executed by one accustomed to such writing. They are, however, so distorted, that some time elapsed before it was even suspected that they were Arabic.