Encyclopaedia Britannica, 11th Edition, "Groups, Theory of" to "Gwyniad" Volume 12, Slice 6

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THE ENCYCLOPÆDIA BRITANNICA

A DICTIONARY OF ARTS, SCIENCES, LITERATURE AND GENERAL INFORMATION

ELEVENTH EDITION

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VOLUME XII SLICE VI

Groups, Theory of to Gwyniad

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Articles in This Slice

GROUPS, THEORY OF GUIDICCIONI, GIOVANNI GROUSE GUIDO OF AREZZO GROVE, SIR GEORGE GUIDO OF SIENA GROVE, SIR WILLIAM ROBERT GUIDO RENI GROVE GUIENNE GROZNYI GUIGNES, JOSEPH DE GRUB GUILBERT, YVETTE GRUBER, JOHANN GOTTFRIED GUILDFORD GRUMBACH, WILHELM VON GUILDHALL GRUMENTUM GUILFORD, BARONS AND EARLS OF GRÜN GUILFORD GRÜNBERG GUILLAUME, JEAN BAPTISTE CLAUDE EUGÈNE GRUNDTVIG, NIKOLAI FREDERIK SEVERIN GUILLAUME DE LORRIS GRUNDY, SYDNEY GUILLAUME DE PALERME GRUNDY, MRS GUILLAUME D’ORANGE GRUNER, GOTTLIEB SIGMUND GUILLEMOT GRÜNEWALD, MATHIAS GUILLOCHE GRUTER, JAN GUILLON, MARIE NICOLAS SYLVESTRE

GRUYÈRE GUILLOTINE GRYNAEUS, JOHANN JAKOB GUILT GRYNAEUS, SIMON GUIMARÃES GRYPHIUS, ANDREAS GUIMARD, MARIE MADELEINE GUACHARO GUIMET, JEAN BAPTISTE GUACO GUINEA (Africa) GUADALAJARA (city of Mexico) GUINEA (gold coin) GUADALAJARA (province of Spain) GUINEA FOWL GUADALAJARA (city of Spain) GUINEA-WORM GUADALQUIVIR GÜINES GUADELOUPE GUINGAMP GUADET, MARGUERITE ÉLIE GUINNESS GUADIANA GUINOBATAN GUADIX GUIPÚZCOA GUADUAS GUIRAUD, ERNEST GUAIACUM GUISBOROUGH GUALDO TADINO GUISE GUALEGUAY GUISE, HOUSE OF GUALEGUAYCHÚ GUITAR GUALO, CARDINAL GUITAR FIDDLE GUAM GUITRY, LUCIEN GERMAIN GUAN

tag="{http://www.w3.org/1999/xhtml}a">GUIZOT, FRANÇOIS PIERRE GUILLAUME GUANABACOA GUJARAT GUANACO GUJARATI and RAJASTHANI GUANAJAY GUJRANWALA GUANAJUATO (state of Mexico) GUJRAT GUANAJUATO (city of Mexico) GULA GUANCHES GULBARGA GUANIDINE GULF STREAM GUANO GULFWEED GUANTA GULL, SIR WILLIAM WITHEY GUANTÁNAMO GULL GUARANA GULLY, JOHN GUARANIS GULPÁÏGÁN GUARANTEE GUM GUARATINGUETÁ GÜMBEL, KARL WILHELM VON GUARDA GUMBINNEN GUARDI, FRANCESCO GUMBO GUARDIAN GUMTI GUARDS, and HOUSEHOLD TROOPS GUMULJINA GUARD-SHIP GUMUS GUÁRICO GÜMÜSH-KHANEH GUARIENTO GUN GUARINI, CAMILLO-GUARINO

tag="{http://www.w3.org/1999/xhtml}a">GUNA GUARINI, GIOVANNI BATTISTA GUNCOTTON GUARINO GUNDULICH, IVAN GUARINO [GUARINUS] DA VERONA GUNG’L, JOSEF GUARNIERI GUNNER GUASTALLA GUNNING, PETER GUATEMALA (republic) GUNNY GUATEMALA (city of Guatemala) GUNPOWDER GUATOS GUNPOWDER PLOT GUATUSOS GUN-ROOM GUAVA GUNTER, EDMUND GUAYAMA GÜNTHER, JOHANN CHRISTIAN GUAYAQUIL GÜNTHER OF SCHWARZBURG GUAYAS GUNTRAM GUAYCURUS GUNTUR GUAYMAS GUPTA GUBBIO GURA, EUGEN GUBEN GURDASPUR GUBERNATIS, ANGELO DE GURGAON GUDBRANDSDAL GURKHA GUDE, MARQUARD GURNALL, WILLIAM GUDEMAN, ALFRED GURNARD GUDGEON

tag="{http://www.w3.org/1999/xhtml}a">GURNEY GUDRUN GURNEY, EDMUND GUÉBRIANT, JEAN BAPTISTE BUDES GURWOOD, JOHN GUELDER ROSE GUSLA GUELPH GUSTAVUS I. ERIKSSON GUELPHS AND GHIBELLINES GUSTAVUS II. ADOLPHUS GUENEVERE GUSTAVUS III. GUENON GUSTAVUS IV. GUÉRET GUSTAVUS V. GUEREZA GUSTAVUS ADOLPHUS UNION GUERICKE, HEINRICH ERNST FERDINAND GÜSTROW GUERICKE, OTTO VON GUTENBERG, JOHANN GUÉRIDON GÜTERSLOH GUÉRIN, JEAN BAPTISTE PAULIN GUTHRIE, SIR JAMES GUÉRIN, PIERRE NARCISSE GUTHRIE, THOMAS GUÉRIN DU CAYLA, GEORGES MAURICE DE GUTHRIE, THOMAS ANSTEY GUERNIERI GUTHRIE GUERNSEY GUTHRUM GUERRAZZI, FRANCESCO DOMENICO GUTSCHMID, ALFRED GUERRERO GUTS-MUTHS, JOHANN CHRISTOPH FRIEDRICH GUERRILLA GUTTA GUERRINI, OLINDO GUTTA PERCHA

tag="{http://www.w3.org/1999/xhtml}a">GUESDE, JULES BASILE GUTTER GUEST, EDWIN GUTZKOW, KARL FERDINAND GUEST GÜTZLAFF, KARL FRIEDRICH AUGUST GUETTARD, JEAN ÉTIENNE GUY OF WARWICK GUEUX, LES GUY, THOMAS GUEVARA, ANTONIO DE GUYON, JEANNE MARIE BOUVIER DE LA MOTHE GUEVARA, LUIS VELEZ DE GUYON, RICHARD DEBAUFRE GUGLIELMI, PIETRO GUYOT, ARNOLD HENRY GUIANA GUYOT, YVES GUIART, GUILLAUME GUYTON DE MORVEAU, LOUIS BERNARD GUIBERT (of Ravenna) GUZMICS, IZIDÓR GUIBERT (of Nogent) GWADAR GUIBERT, JACQUES ANTOINE HIPPOLYTE GWALIOR GUICCIARDINI, FRANCESCO GWEEDORE GUICHARD, KARL GOTTLIEB GWILT, JOSEPH GUICHEN, LUC URBAIN DE BOUËXIC GWYN, NELL GUIDE GWYNIAD GUIDI, CARLO ALESSANDRO  

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GROUPS,1 THEORY OF. The conception of an operation to be carried out on some object or set of objects underlies all mathematical science. Thus in elementary arithmetic there are the fundamental operations of the addition and the multiplication of integers; in algebra a linear transformation is an operation which may be carried out on any set of variables; while in geometry a translation, a rotation, or a projective transformation are operations which may be carried out on any figure.

In speaking of an operation, an object or a set of objects to which it may be applied is postulated; and the operation may, and generally will, have no meaning except in regard to such a set of objects. If two operations, which can be performed on the same set of objects, are such that, when carried out in succession on any possible object, the result, whichever operation is performed first, is to produce no change in the object, then each of the operations is spoken of as a definite operation, and each of them is called the inverse of the other. Thus the operations which consist in replacing x by nx and by x/n respectively, in any rational function of x, are definite inverse operations, if n is any assigned

number except zero. On the contrary, the operation of replacing x by an assigned number in any rational function of x is not, in the present sense, although it leads to a unique result, a definite operation; there is in fact no unique inverse operation corresponding to it. It is to be noticed that the question whether an operation is a definite operation or no may depend on the range of the objects on which it operates. For example, the operations of squaring and extracting the square root are definite inverse operations if the objects are restricted to be real positive numbers, but not otherwise.