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
GRUN GUILFORD
GRUNBERG GUILLAUME, JEAN BAPTISTE CLAUDE EUGENE
GRUNDTVIG, NIKOLAI SEVERIN GUILLAUME DE LORRIS
GRUNDY, SYDNEY GUILLAUME DE PALERME
GRUNDY, MRS GUILLAUME D'ORANGE
GRUNER, GOTTLIEB SIGMUND GUILLEMOT
GRUNEWALD, MATHIAS GUILLOCHE
GRUTER, JAN GUILLON, MARIE NICOLAS SYLVESTRE
GRUYERE GUILLOTINE
GRYNAEUS, JOHANN JAKOB GUILT
GRYNAEUS, SIMON GUIMARAES
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 GUINES
GUADELOUPE GUINGAMP
GUADET, MARGUERITE ELIE GUINNESS
GUADIANA GUINOBATAN
GUADIX GUIPUZCOA
GUADUAS GUIRAUD, ERNEST
GUAIACUM GUISBOROUGH
GUALDO TADINO GUISE
GUALEGUAY GUISE, HOUSE OF
GUALEGUAYCHU GUITAR
GUALO, CARDINAL GUITAR FIDDLE
GUAM GUITRY, LUCIEN GERMAIN
GUAN GUIZOT, FRANCOIS 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
GUANTANAMO GULL
GUARANA GULLY, JOHN
GUARANIS GULPAIGAN
GUARANTEE GUM
GUARATINGUETA GUMBEL, KARL WILHELM VON
GUARDA GUMBINNEN
GUARDI, FRANCESCO GUMBO
GUARDIAN GUMTI
GUARDS, and HOUSEHOLD TROOPS GUMULJINA
GUARD-SHIP GUMUS
GUARICO GUMUSH-KHANEH
GUARIENTO GUN
GUARINI, CAMILLO-GUARINO 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 GUNTHER, JOHANN CHRISTIAN
GUAYAQUIL GUNTHER 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 GURNEY
GUDRUN GURNEY, EDMUND
GUEBRIANT, JEAN BAPTISTE BUDES GURWOOD, JOHN
GUELDER ROSE GUSLA
GUELPH GUSTAVUS I. ERIKSSON
GUELPHS AND GHIBELLINES GUSTAVUS II. ADOLPHUS
GUENEVERE GUSTAVUS III.
GUENON GUSTAVUS IV.
GUERET GUSTAVUS V.
GUEREZA GUSTAVUS ADOLPHUS UNION
GUERICKE, HEINRICH FERDINAND GUSTROW
GUERICKE, OTTO VON GUTENBERG, JOHANN
GUERIDON GUTERSLOH
GUERIN, JEAN BAPTISTE PAULIN GUTHRIE, SIR JAMES
GUERIN, PIERRE NARCISSE GUTHRIE, THOMAS
GUERIN DU CAYLA, 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
GUESDE, JULES BASILE GUTTER
GUEST, EDWIN GUTZKOW, KARL FERDINAND
GUEST GUTZLAFF, KARL FRIEDRICH AUGUST
GUETTARD, JEAN ETIENNE GUY OF WARWICK
GUEUX, LES GUY, THOMAS
GUEVARA, ANTONIO DE GUYON, JEANNE 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, IZIDOR
GUIBERT (of Nogent) GWADAR
GUIBERT, JACQUES HIPPOLYTE GWALIOR
GUICCIARDINI, FRANCESCO GWEEDORE
GUICHARD, KARL GOTTLIEB GWILT, JOSEPH
GUICHEN, LUC URBAIN DE BOUEXIC GWYN, NELL
GUIDE GWYNIAD
GUIDI, CARLO ALESSANDRO
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.
If O, O', O", ... is the totality of the objects on which a definite
operation S and its inverse S' may be carried out, and if the result
of carrying out S on O is represented by O.S, then O.S.S', O.S'.S, and
O are the same object whatever object of the set O may be. This will
be represented by the equations SS' = S'S = 1. Now O.S.S' has a
meaning only if O.S is an object on which S' may be performed. Hence
whatever object of the set O may be, both O.S and O.S' belong to the
set. Similarly O.S.S, O.S.S.S, ... are objects of the set. These will
be represented by O.S^2, O.S^3, ... Suppose now that T is another
definite operation with the same set of objects as S, and that T' is
its inverse operation. Then O.S.T is a definite operation of the set,
and therefore the result of carrying out S and then T on the set of
objects is some operation U with a unique result. Represent by U' the
result of carrying out T' and then S'. Then O.UU' = O.S.T.T'.S' =
O.SS' = O, and O.U'U = O.T'.S'.S.T = O.T'T = O, whatever object O may
be. Hence UU' = U'U = 1; and U, U' are definite inverse operations.
If S, U, V are definite operations, and if S' is the inverse of S,
then