752 SCHOOL SCIENCE AND MATHEMATICS

GROUP THEORY FOR TEACHERS OF ELEMENTARY MATHE-MATICS.

BY G. A. MILLER,

University of Illinois^ Urbana.

At the fifth annual meeting of the association of the teachersof mathematics in the secondary schools of Switzerland, heldat Zurich on December, 9, 1905, Professor H. Fehr of the Uni-versity of Geneva gave a brief sketch of some present tendenciesin the teaching of elementary geometry. He laid special stresson the fact that the teacher should be familiar with the notion ofa group, which plays such a fertile role at the present time inthe mathematical developments.*The present article aims to exhibit some applications of the

group concept to elementary mathematics. It is hoped that thearticle may serve as an easy introduction to some elements of thegroup theory and also as .a means to arrive at broader notions inreference to^a few fundamental matters in elementary mathe-matics.We shall first consider the two operations of subtracting from

I and dividing i. The former operation will be represented by

^ and the latter by d. The two successive operations of firstsubtracting from i and then dividing i by this remainder will bedenoted by sd. It is easy to prove that we arrive at the originalnumber by performing sd three times in’ succession on any num-ber. Por instance, if we start with 2 and apply sd on it we arriveat �i; applying sd on �i, ,we arrive at ^2; applying sd on ^2,we arrive at the original number 2. In general, the followingthree numbers result if sd is applied successively on the number

n.1�nn

The three numbers n. ���» ��� are distinct except when n1 �n n

has one of the following two values, ^– il/^. Hence the opera-tion sd combines all the numbers (real and imaginary) into setsof threes, with the exception of the two numbers just mentioned.If one number of such a set is real all of them are real, arid ifone is rational all are rational, since the operations of division and

*!/ ^nseignment Mathematique, vol. 8 (1906), p. 54.

GROUP THEORY IN MATHEMATICS 753

subtraction performed on rational numbers lead to lational num-bers.

If we consider the six operations

1 sd (sd)2d ,s sd s (A)

where I as usual represents the identical operation (that is, anoperation which leaves all things unchanged), we can readilyverify that they, in general, associate six distinct numbers. Forinstance, if we perform these six operations on the number nwe obtain

1 n �11 � n n

1 , n.� 1� n ���-n n � 1

One of the most important facts has not yet been explicitly-stated ; viz., that we ’arrive at the same set of numbers by apply-ing all these operations to any one of the set. For instance, ifwe start with 3, the given six operations give rise to the followingnumbers in order: 3,y-, |-, i,’� 2, |. By starting with�i-the order is: �i-, -j-, 3, � 2, f, -1-. Similarly, by starting withany other number of this set and applying to it the six givenoperations there would result merely a rearrangement of thesesix numbers. This is directly due to the fact that the six oper-ations denoted by (A) form a group.

Suppose that these operations are applied in order to sm\v.The resultingfunctions will be: snAf, sec2^,�cot2^,�tan2^, csc2^,cos2^. Hence these six functions represent a set of numbers un-der (A) for every value of x. The numbers of such a set aresaid to be conjugate under the group represented by (A). Whileit would be easy to exhibit many more interesting relations, yetthese are of especial value on account of the fact that they con-stitute a part of a very large body of closely related facts. Thatis, they are simply steps toward grander views.

It has been observed that each of the two numbers -t–i- v/=3is transformed into itself by the first three operations of (A).It is easy to verify that the remainder of these operations trans-form one of these numbers into the other. Hence each of thesenumbers has only two conjugates under (A). Each of the two

*Quarterly Journal of Mathematics, vol. 37 (1905), p. 80.

754 SCHOOL S-CIENCE AND MATHEMATICS

tripl’ets (�1, 2, i), (1, 0, <^) has three conjugates under (A).Every other number has six distinct conjugates under (A). This

statement can readily be proved* and evidently includes questionsrelating to the equality of the trigonometric functions of the pre-ceding paragraph.The operations of (A) can be placed in a (i, i) correspondence

with the six movements which transform an equilateraltriangle into itself. Those of the first line correspond to themovements through o°, 120°, and 240° respectively around thecenter of the triangle, while the remaining three correspond to

movements of the plane of the triangle through 180° around tlielines of symmetry. As these six movements and the opera-tions of (A) obey the same laws of combination they are saidto represent the same abstract group.

Just as the operations of subtracting from i and dividing i

give rise to a group of order 6 so the operations of subtractingfrom 2 or 3 and dividing 2 or 3 give rise to a group of order 8or 12 respectively. In the former case there are 10 special num-

bers which are ^qual to some of their conjugates while in thelatter case there are 14 such numbers. According to these groupsthe other numbers are associated in sets of 8 or 12 respectivelysuch that each number of a set is transformed into every othernumber of the set by means of the operations of the respectivegroups. For details and proofs we refer to the article mentionedin the preceding footnote.

If we consider the eight movements of the plane which trans-form a square into itself, it will readily follow that they can beplaced in (i, i) correspondence with the eight operations ob-tained by combining dz and ^2 where d^ represents dividing 2 and^2 represents subtracting from 2. The operation of ^2 Js has to berepeated four times in order to arrive at the identity and maycorrespond to a rotation through 90° in the group of movementsof the square. That ^2 d^ is generally of period four follows from

.1 ,. , 2 / ^0 2 ���

^ / 7 \q 2%���2the equations: s^d^ =_��? (^9^2) � ���’ (s^d^) = �- )

2 � n 1 � n ^ ^ n

^d^^n.This group of order 8 is generally known as the octic group

and presents itself in a large number of places in elementary

mahematics. We have already noted an instance in arithmetic�subtracting from 2 and dividing 2� and an instance from

GROUP THEORY IN MATHEMATICS 755

elementary geometry�the movements which transform a squareinto itself. We proceed to give a fundamental application of this

group in trigonometry.Denoting the operations of taking the complement and the

supplement of an angle by c and s respectively, it is easy to see

that cs increases the angle by 90°. Hence (^.O^^ISO0,{es)^= a + 270°, (cs)4:= a, where a is the angle under consider-ation. From this it is clear that cs is an operation of periodfour just as ^2 d^. Moreover, the eight operations

1 cs (cs)2 (^)3s, c csc scs

form a group which has exactly the same properties as the groupgenerated by sz da.

It is especially interesting to observe that the eight operationsof the octis group transform a into. the eight angles whose func-tions are generally tabulated in the elementary text-books on

trigonometry. These angles are in order: a, a -{- 90° a +180°,a + 270°, 180°� a, 90°� a, � a, 270° � a. On account of theprominence of these angles and of the operations c and s ourelementary trigonometry might appropriately be called the trig-onometry of the octic group. Some of the methods of employingthe properties of the octic group in the study of the functionsof these eight angles have been given in the article entitled ’ ’Anew chapter in trigonometry," Quarterly Journal of Mathe-matice, vol. 37, (1906), p. 226.What precedes, exhibits, some applications of the group of the

triangle and the square in other elementary, subjects and thus ex-hibits an interesting relation betwen these subjects. This isone of the most important features of the group concept. It isnot implied that these relations should be made prominent in ele-mentary instruction but the teacher who knows them will teachmore wisely and with a deeper interest than if he were ignorantof them. What is needed is a clear understanding of the prin-ciples which find extensive applications. Isolated facts are fre-quently of interest but they cannot be as fruitful as those whichfind extensive applications in further developments. Only thereal scholar can be a judge of the relative importance of theelements which enter into a mathematical training.The three regular polygons which enter most extensively into

the study of elementary geometry ’are the triangle, the square,and the hexagon. We have briefly considered the group of

756 , SCHOOL SCIENCE AND MATHEMATICS

movements of the first two. The group of movements of the lastis of order 12 and has the same properties as the group generatedby the two operations of .subtracting from 3 and dividing 3. Ifthese two operations are represented by ss and d3 respectively,it is easy to verify that S3 d3 is an operation of period six, whichmay correspond to a rotation through 60°. The other operationsof the group generated by ss, ds correspond to rotations aroundthe lines of symmetry of the regular hexagon.

It is a curious fact that the groups of the three fundamentalregular polygons should be the same as the three finite groups ofsubtraction and division, with rational numbers, if we excludethe almost trivial case of subtracting from o and dividing i.

While each of the other regular polygons has a group of move-ments whose order is also twice the number of sides of thepolygon, yet these groups do not present themselves as subtrac-tion and division groups when the numbers from which we sub-tract and which is divided are both rational.* In view of thesefacts the groups which have been considered are of somewhatspecial interest.The principal aims of a brief article on a big subject should

be to arouse a healthy interest and to give suitable references.In trying to supply the latter we would especially refer to thearticle by Poincare published in The Monist, vol. 9 (1898, p. 34.Similar views were employed as early as 1874 in Meray^s Nou-veux Elements de Geometric, For briefer developments alongthis line we may refer to the definition of group theory in ThePopular Science Monthly, February, 1904, "on the groups of thefigures of elementary geometry," American MathematicalMonthly, October, 1903, and to the articles mentioned above.

SOME THOUGHTS ON THE TEACHING OF GEOMETRY.

C. A. PETTERSEN.Jefferson High School, Chicago.

Not many years ago there was graduated ’from a universitynot far distant from Chicago a young man who was thus de-scribed in a "grind" in the College Annual:

"A sober youth with solemn phiz .

Who eats his grub and minds his biz."Whether the author of this jingle was aware of the destiny of

*Cf. Hilton, Messeng-er of Mathematics, vol. 35 (1905), p. 117.