Part 1

Produced by Marius Masi, Don Kretz and the Online Distributed Proofreading Team at http://www.pgdp.net

Transcriber's notes:

(1) Numbers following letters (without space) like C2 were originally printed in subscript. Letter subscripts are preceded by an underscore, like C_n.

(2) Characters following a carat (^) were printed in superscript.

(3) Side-notes were relocated to function as titles of their respective paragraphs.

(4) Macrons and breves above letters and dots below letters were not inserted.

(5) [root] stands for the root symbol; [alpha], [beta], etc. for greek letters and [Pd] for partial differential symbol.

(6) The following typographical errors have been corrected:

ARTICLE GUADALQUIVIR: "Here it forms two subsidiary channels, the western 31 m., the eastern 12 m. long, which rejoin the main stream on the borders of the province of Cadiz." 'm.' amended from 'M.'.

ARTICLE GUANAJUATO: "... W. of Guanajuato in a rich mining district; and Acambaro (8345), a prosperous town of the plain, 76 m. S.S.E. of Guanajuato." 'Guanajuato' amended from 'Guanaiuato'.

ARTICLE GUARANTEE: "The Egyptian codes sanction guarantees expressly entered into 'in view of debtor's want of legal capacity' to contract a valid principal obligation (Egyptian Codes, Mixed Suits, 605; Native Tribunals, 496)." 'Egyptian' amended from 'Egyptain'.

ARTICLE GUINEA FOWL: "Allied to the genus Numida, but readily distinguished form among other characters by the possession of spurs and the absence of a helmet, are two very rare forms ..." 'form' amended from 'thereform'.

ARTICLE GUIPUZCOA: "The principal industrial centres are Irun, Renteria, Villabona, Vergara and Azpeitia for cotton and linen stuffs; Zumarraga for osiers; Eibar, Plasencia and Elgoibar for arms and cannon and gold incrustations; ..." 'osiers' amended from 'osies'.

ARTICLE GUTZKOW, KARL FERDINAND: "The success of Die Ritter vom Geiste suggested to Gutzkow the establishment of a journal on the model of Dickens' Household Words, entitled Unterhaltungen am hauslichen Herd, which first appeared in 1852 and was continued till 1862." "Dickens'" amended from "Dicken's".

ARTICLE GUY OF WARWICK: "... The Tragical History, Admirable Achievements and Curious Events of Guy, Earl of Warwick ..." 'Achievements' amended from 'Atchievements'.

ENCYCLOPAEDIA BRITANNICA

A DICTIONARY OF ARTS, SCIENCES, LITERATURE AND GENERAL INFORMATION

ELEVENTH EDITION

VOLUME XII, SLICE VI

Groups, Theory of to Gwyniad

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 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

SU = SV

implies S'SU = S'SV,

or U = V.

Similarly US = VS

implies U = V.

Definition of a group.

Let S, T, U, ... be a set of definite operations, capable of being carried out on a common object or set of objects, and let the set contain--

(i.) the operation ST, S and T being any two operations of the set;

(ii.) the inverse operation of S, S being any operation of the set; the set of operations is then called a group.

The number of operations in a group may be either finite or infinite. When it is finite, the number is called the _order_ of the group, and the group is spoken of as a _group of finite order_. If the number of operations is infinite, there are three possible cases. When the group is represented by a set of geometrical operations, for the specification of an individual operation a number of measurements will be necessary. In more analytical language, each operation will be specified by the values of a set of parameters. If no one of these parameters is capable of continuous variation, the group is called a _discontinuous group_. If all the parameters are capable of continuous variation, the group is called a _continuous group_. If some of the parameters are capable of continuous variation and some are not, the group is called a _mixed group_.

If S' is the inverse operation of S, a group which contains S must contain SS', which produces no change on any possible object. This is called the _identical operation_, and will always be represented by I. Since S^pS^q = S^(p+q) when p and q are positive integers, and S^pS' = S^(p-1) while no meaning at present has been attached to S^q when q is negative, S' may be consistently represented by S^(-1). The set of operations ..., S^(-2), S(-1), 1, S, S^2, ... obviously constitute a group. Such a group is called a _cyclical_ group.

Subgroups, conjugate operations, isomorphism, &c.

It will be convenient, before giving some illustrations of the general group idea, to add a number of further definitions and explanations which apply to all groups alike. If from among the set of operations S, T, U, ... which constitute a group G, a smaller set S', T', U', ... can be chosen which themselves constitute a group H, the group H is called a _subgroup_ of G. Thus, in particular, if S is an operation of G, the cyclical group constituted by ..., S^(-2), S^(-1), 1, S, S^2, ... is a subgroup of G, except in the special case when it coincides with G itself.

If S and T are any two operations of G, the two operations S and T^(-1)ST are called _conjugate_ operations, and T^(-1)ST is spoken of as the result of _transforming_ S by T. It is to be noted that since ST = T^(-1), TS, T, ST and TS are always conjugate operations in any group containing both S and T. If T transforms S into itself, that is, if S = T^(-1)ST or TS = ST, S and T are called _permutable_ operations. A group whose operations are all permutable with each other is called an _Abelian_ group. If S is transformed into itself by every operation of G, or, in other words, if it is permutable with every operation of G, it is called a _self-conjugate_ operation of G.

The conception of operations being conjugate to each other is extended to subgroups. If S', T', U', ... are the operations of a subgroup H, and if R is any operation of G, then the operations R^(-1)S'R, R^(-1)T'R, R^(-1)U'R, ... belong to G, and constitute a subgroup of G. For if S'T' = U', then R^(-1)S'R.R^(-1)T'R = R^(-1)S'T'R = R^(-1)U'R. This subgroup may be identical with H. In particular, it is necessarily the same as H if R belongs to H. If it is not identical with H, it is said to be _conjugate_ to H; and it is in any case represented by the symbol R^(-1)HR. If H = R^(-1)HR, the operation R is said to be permutable with the subgroup H. (It is to be noticed that this does not imply that R is permutable with each operation of H.)

If H = R^(-1)HR, when for R is taken in turn each of the operations of G, then H is called a _self-conjugate_ subgroup of G.

A group is spoken of as _simple_ when it has no self-conjugate subgroup other than that constituted by the identical operation alone. A group which has a self-conjugate subgroup is called _composite_.

Let G be a group constituted of the operations S, T, U, ..., and g a second group constituted of s, t, u, ..., and suppose that to each operation of G there corresponds a single operation of g in such a way that if ST = U, then _st_ = u, where s, t, u are the operations corresponding to S, T, U respectively. The groups are then said to be _isomorphic_, and the correspondence between their operations is spoken of as an _isomorphism_ between the groups. It is clear that there may be two distinct cases of such isomorphism. To a single operation of g there may correspond either a single operation of G or more than one. In the first case the isomorphism is spoken of as _simple_, in the second as _multiple_.

Two simply isomorphic groups considered abstractly--that is to say, in regard only to the way in which their operations combine among themselves, and apart from any concrete representation of the operations--are clearly indistinguishable.

If G is multiply isomorphic with g, let A, B, C, ... be the operations of G which correspond to the identical operation of g. Then to the operations A^(-1) and AB of G there corresponds the identical operation of g; so that A, B, C, ... constitute a subgroup H of G. Moreover, if R is any operation of G, the identical operation of g corresponds to every operation of R^(-1)HR, and therefore H is a self-conjugate subgroup of G. Since S corresponds to s, and every operation of H to the identical operation of g, therefore every operation of the set SA, SB, SC, ..., which is represented by SH, corresponds to s. Also these are the only operations that correspond to s. The operations of G may therefore be divided into sets, no two of which contain a common operation, such that the correspondence between the operations of G and g connects each of the sets H, SH, TH, UH, ... with the single operations 1, s, t, u, ... written below them. The sets into which the operations of G are thus divided combine among themselves by exactly the same laws as the operations of g. For if st = u, then SH.TH = UH, in the sense that any operation of the set SH followed by any operation of the set TH gives an operation of the set UH.

The group g, abstractly considered, is therefore completely defined by the division of the operations of G into sets in respect of the self-conjugate subgroup H. From this point of view it is spoken of as the _factor-group_ of G in respect of H, and is represented by the symbol G/H. Any composite group in a similar way defines abstractly a factor-group in respect of each of its self-conjugate subgroups.

It follows from the definition of a group that it must always be possible to choose from its operations a set such that every operation of the group can be obtained by combining the operations of the set and their inverses. If the set is such that no one of the operations belonging to it can be represented in terms of the others, it is called a set of _independent generating_ operations. Such a set of generating operations may be either finite or infinite in number. If A, B, ..., E are the generating operations of a group, the group generated by them is represented by the symbol {A, B, ..., E}. An obvious extension of this symbol is used such that {A, H} represents the group generated by combining an operation A with every operation of a group H; {H1, H2} represents the group obtained by combining in all possible ways the operations of the groups H1 and H2; and so on. The independent generating operations of a group may be subject to certain relations connecting them, but these must be such that it is impossible by combining them to obtain a relation expressing one operation in terms of the others. For instance, AB = BA is a relation conditioning the group {A, B}; it does not, however, enable A to be expressed in terms of B, so that A and B are independent generating operations.

Transitivity and primitivity.

Let O, O', O", ... be a set of objects which are interchanged among themselves by the operations of a group G, so that if S is any operation of the group, and O any one of the objects, then O.S is an object occurring in the set. If it is possible to find an operation S of the group such that O.S is any assigned one of the set of objects, the group is called _transitive_ in respect of this set of objects. When this is not possible the group is called _intransitive_ in respect of the set. If it is possible to find S so that any arbitrarily chosen n objects of the set, O1, O2, ..., O_n are changed by S into O'1, O'2, ..., O'n respectively, the latter being also arbitrarily chosen, the group is said to be n-ply transitive.

If O, O', O", ... is a set of objects in respect of which a group G is transitive, it may be possible to divide the set into a number of subsets, no two of which contain a common object, such that every operation of the group either interchanges the objects of a subset among themselves, or changes them all into the objects of some other subset. When this is the case the group is called _imprimitive_ in respect of the set; otherwise the group is called _primitive_. A group which is doubly-transitive, in respect of a set of objects, obviously cannot be imprimitive.

Illustrations of the group idea.

The foregoing general definitions and explanations will now be illustrated by a consideration of certain particular groups. To begin with, as the operations involved are of the most familiar nature, the group of rational arithmetic may be considered. The fundamental operations of elementary arithmetic consist in the addition and subtraction of integers, and multiplication and division by integers, division by zero alone omitted. Multiplication by zero is not a definite operation, and it must therefore be omitted in dealing with those operations of elementary arithmetic which form a group. The operation that results from carrying out additions, subtractions, multiplications and divisions, of and by integers a finite number of times, is represented by the relation x' = ax + b, where a and b are rational numbers of which a is not zero, x is the object of the operation, and x' is the result. The totality of operations of this form obviously constitutes a group.

If S and T represent respectively the operations x' = ax + b and x' = cx + d, then T^(-1)ST represents x' = ax + d - ad + bc. When a and b are given rational numbers, c and d may be chosen in an infinite number of ways as rational numbers, so that d - ad + bc shall be any assigned rational number. Hence the operations given by x' = ax + b, where a is an assigned rational number and b is any rational number, are all conjugate; and no two such operations for which the a's are different can be conjugate. If a is unity and b zero, S is the identical operation which is necessarily self-conjugate. If a is unity and b different from zero, the operation x' = x + b is an addition. The totality of additions forms, therefore, a single conjugate set of operations. Moreover, the totality of additions with the identical operation, i.e. the totality of operations of the form x' = x + b, where b may be any rational number or zero, obviously constitutes a group. The operations of this group are interchanged among themselves when transformed by any operation of the original group. It is therefore a self-conjugate subgroup of the original group.

The totality of multiplications, with the identical operation, i.e. all operations of the form x' = ax, where a is any rational number other than zero, again obviously constitutes a group. This, however, is not a self-conjugate subgroup of the original group. In fact, if the operations x' = ax are all transformed by x' = cx + d, they give rise to the set x' = ax + d(1 - a). When d is a given rational number, the set constitutes a subgroup which is conjugate to the group of multiplications. It is to be noticed that the operations of this latter subgroup may be written in the form x' - d = a(x - d).

The totality of rational numbers, including zero, forms a set of objects which are interchanged among themselves by all operations of the group.

If x1 and x2 are any pair of distinct rational numbers, and y1 and y2 any other pair, there is just one operation of the group which changes x1 and x2 into y1 and y2 respectively. For the equations y1 = ax1 + b, y1 = ax2 + b determine a and b uniquely. The group is therefore doubly transitive in respect of the set of rational numbers. If H is the subgroup that leaves unchanged a given rational number x1, and S an operation changing x1 into x2, then every operation of S^(-1)HS leaves x2 unchanged. The subgroups, each of which leaves a single rational number unchanged, therefore form a single conjugate set. The group of multiplications leaves zero unchanged; and, as has been seen, this is conjugate with the subgroup formed of all operations x' - d = a(x - d), where d is a given rational number. This subgroup leaves d unchanged.

The group of multiplications is clearly generated by the operations x' = px, where for p negative unity and each prime is taken in turn. Every addition is obtained on transforming x' = x + 1 by the different operations of the group of multiplications. Hence x' = x + 1, and x' = px, (p = -1, 3, 5, 7, ...), form a set of independent generating operations of the group. It is a discontinuous group.