Groups of the Order p^m

Lewis Irving Neikirk

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Title: Groups of Order p^m Which Contain Cyclic Subgroups of Order p^(m-3) Author: Lewis Irving Neikirk Release Date: February, 2006 [EBook #9930] [Yes, we are more than one year ahead of schedule] [This file was first posted on November 1, 2003] Edition: 10 Language: English Character set encoding: TeX *** START OF THE PROJECT GUTENBERG EBOOK GROUPS OF ORDER P^M ***

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GROUPS OF ORDERp m WHICH CONTAIN CYCLIC SUBGROUPS OF ORDERp m−3

by

LEWIS IRVING NEIKIRK sometime harrison research fellow in mathematics

1905

2

INTRODUCTORY NOTE. This monograph was begun in 1902-3. Class I, Class II, Part I, and the self- conjugate groups of Class III, which contain all the groups with independent generators, formed the thesis which I presented to the Faculty of Philosophy of the University of Pennsylvania in June, 1903, in partial fulfillment of the requirements for the degree of Doctor of Philosophy. The entire paper was rewritten and the other groups added while the author was Research Fellow in Mathematics at the University. I wish to express here my appreciation of the opportunity for scientific re- search afforded by the Fellowships on the George Leib Harrison Foundation at the University of Pennsylvania. I also wish to express my gratitude to Professor George H. Hallett for his kind assistance and advice in the preparation of this paper, and especially to express my indebtedness to Professor Edwin S. Crawley for his support and encouragement, without which this paper would have been impossible.

University Of Pennsylvania,May, 1905.

3

Lewis I. Neikirk.

GROUPS OF ORDERp m , WHICH CONTAIN CYCLIC SUBGROUPS OF ORDERp (m−3)1 by lewis irving neikirk Introduction. The groups of orderp m , which contain self-conjugate cyclic subgroups of ordersp m−1 , andp m−2 respectively, have been determined byBurnside, 2 and the number of groups of orderp m , which contain cyclic non-self-conjugate sub- groups of orderp m−2 has been given byMiller. 3 Although in the present state of the theory, the actual tabulation of all groups of orderp m is impracticable, it is of importance to carry the tabulation as far as may be possible. In this paperall groups of orderp m (pbeing an odd prime)which contain cyclic subgroups of orderp m−3 and none of higher order are determined. The method of treatment used is entirely abstract in character and, in virtue of its nature, it is possible in each case to give explicitly the generational equations of these groups. They are divided into three classes, and it will be shown that these classes correspond to the three partitions: (m−3,3), (m−3,2,1) and (m−3,1,1,1), ofm. We denote byGan abstract groupGof orderp m containing operators of orderp m−3 no operator of order greater thanand p m−3 . LetPdenote one of these operators ofGof orderp m−3 . Thep 3 power of every operator inGis contained in the cyclic subgroup{P}, otherwiseGwould be of order greater than p m division into classes is effected by the following assumptions: complete . The I. There is inGat least one operatorQ 1 , such thatQ p 1 2 is not contained in {P}. II. Thep 2 power of every operator inGis contained in{P}, and there is at least one operatorQ 1 , such thatQ 1 p is not contained in{P}. III. Thepth power of every operator inGis contained in{P}.

1 Presented to the American Mathematical Society April 25, 1903. 2 Theory of Groups of a Finite Order, pp. 75-81. 3 Transactions, vol. 2 (1901), p. 259, and vol. 3 (1902), p. 383. 4

The number of groups for Class I, Class II, and Class III, together with the total number, are given in the table below: I II 1 II 2 II 3 II III Total p >3 m > +8 9 20p6 + 2p6 + 2p32 + 5p23 64 + 5p p >3 m= 8 + 8 20p6 + 2p6 + 2p32 + 5p23 63 + 5p p >3 m 6 20 += 7p6 + 2p6 + 2p32 + 5p23 61 + 5p p= 3 m >8 9 23 12 12 47 16 72 p= 3 m 47 16 71 23 12 12= 8 8 p= 3 m 47 12 12 23 6 16 69= 7

ClassI. 1.General notations and relations.—The groupGis generated by the two operatorsPandQ 1  brevity we set. For 4 Q a 1 P b Q c 1 P d ∙ ∙ ∙= [a, b, c, d,∙ ∙ ∙]. Then the operators ofGare given each uniquely in the form [y, x]
x y0 =0 = , ,1 1, ,2 2, ,∙ ∙∙ ∙∙ ∙ p ,p , 3 m − −3 1 −1 ! . We have the relation (1)Q p 1 3 =P hp 3 . There is inG, a subgroupH 1 of orderp m−2 , which contains{P}self-conjugate- ly. 5 The subgroupH 1 is generated byPand some operatorQ y 1 P x ofG; it then 2 containsQ y 1 and is therefore generated byPandQ 1 p ; it is also self-conjugate inH 2 ={Q p 1 , P}of orderp m−1 , andH 2 is self-conjugate inG. From these considerations we have the equations 6 (2)Q 1 −p 2 P Q 1 p 2 =P 1+kp m−4 , Q −p P Q 1 p =Q βp 2 (3) 1 1 P α 1 , (4)Q 1 −1 P Q 1 =Q b 1 p P a 1 . 4 With J. W.Young,On a certain group of isomorphisms, American Journal of Mathe- matics, vol. 25 (1903), p. 206. 5 Burnside:Theory of Groups, Art. 54, p. 64. 6 Ibid., Art. 56, p. 66. 5

2.Determination ofH 1  of a formula for. Derivation[yp 2 , x] s .—From (2), by repeated multiplication we obtain [−p 2 , x, p 2 ] = [0, x(1 +kp m−4 )]; and by a continued use of this equation we have [−yp 2 , x, yp 2 ] = [0, x(1 +kp m−4 ) y ] = [0, x(1 +kyp m−4 )] (m >4) and from this last equation, (5) [yp 2 , x] s =  syp 2 , x{s+k  s  yp m−4 }  . 2 3.Determination ofH 2  of a formula for. Derivation[yp, x] s .—It follows from (3) and (5) that [−p 2 ,1, p 2 ] =  β α 1 p 1 − − 1 1 p 2 , α p 1  1 + β 2 αk α 1 p 1 − − 1 1 p m−4  (m >4). α Hence, by (2), β α p 1 −1 p 2 m α 1 −1 ≡0 ( odp 3 ), α p 1  1 + β 2 αk α p 1 1 − − 1 1 p m−4  + α β α 1 p 1 − − 1 1 hp 2 ≡1 +kp m−4 (modp m− ). 3 From these congruences, we have form >6 α 1 p ≡1 (modp 3 ), α 1 ≡1 (modp 2 ), and obtain, by setting 2 α 1 = 1 +α 2 p , the congruence (1 +α α 2 p p 2 3 ) p − ( 1 α 2 +hβ)p 3 ≡kp m−4 (modp m−3 ); 2 and so since

(α 2 +hβ)p 3 ≡0 (modp m−4 ),

+α p −1 (1 α 2 2 p p 3 2 ) ≡1 (modp 2 ). 6

From the last congruences (6) (α 2 +hβ)p 3 ≡kp m−4 (modp m−3 ). Equation (3) is now replaced by (7)Q 1 −p P Q 1 −p =Q 1 βp 2 P 1+α 2 p 2 . From (7), (5), and (6) [−yp, x, yp] =  βxyp 2 , x{1 +α 2 yp 2 }+βk  2 x  yp m−4  . A continued use of this equation gives (8) [yp, x] s = [syp+β  2 s  xyp 2 , xs+  s 2  {α 2 xyp 2 +βk  x 2  yp m−4 }+βk  3 s  x 2 yp m−4 ]. 4.Determination ofG.—From (4) and (8), [−p,1, p] = [N p, a p 1 +M p 2 ]. From the above equation and (7), a p 1 ≡1 (modp 2 ), a 1 ≡1 (modp). Seta 1 = 1 +a 2 pand equation (4) becomes (9)Q 1 −1 P Q 1 =Q 1 bp P 1+a 2 p . From (9), (8) and (6) [−p 2 ,1, p 2 ] = " (1 +a a 2 2 p) p p 2 −1 bp,(1 +a 2 p) p 2 # , and from (1) and (2) (1 +a a 2 2 p p ) p 2 −1 bp≡0 (modp 3 ), (1 +a 2 p) p 2 +bh (1 +a a 2 2 p p) p 2 −1 p≡1 +kp m−4 (modp m−3 ). By a reduction similar to that used before, (10) (a 2 +bh)p 3 ≡kp m−4 (modp m−3 ). The groups in this class are completely defined by (9), (1) and (10). 7

These defining relations may be presented in simpler form by a suitable choice of the second generatorQ 1 . From (9), (6), (8) and (10) [1, x] p 3 = [p 3 , xp 3 ] = [0,(x+h)p 3 ] (m >6), and, ifxbe so chosen that x+h≡0 (modp m−6 ), Q 1 P x is an operator of orderp 3 whosep 2 power is not contained in{P}. Let Q 1 P x =Q. The groupGis generated byQandP, where 3 Q p 3 = 1, P p m− = 1 . Placingh= 0 in (6) and (10) we find α 2 p 3 ≡a 2 p 3 ≡kp m−4 (modp m−3 ). Letα 2 =αp m−7 , anda 2 =ap m−7  (7) and (9) are now replaced by. Equations (11) Q −p P Q p =Q βp 2 P 1+αp m−5 , Q −1 P Q=Q bp P 1+ap m−6 . As a direct result of the foregoing relations, the groups in this class corre- spond to the partition (m−3, (11) we find3). From 7 [−y,1, y] = [byp,1 +ayp m−6 ] (m >8). It is important to notice that by placingy=pandp 2 in the preceding equation we find that 8 b≡β(modp), a≡α≡k(modp 3 ) (m >7). A combination of the last equation with (8) yields 9 (12) [−y, x, y] = [bxyp+b 2  x 2  yp 2 , x(1 +ayp m−6 ) +ab  x 2  yp m−5 +ab 2  x 3  yp m−4 ] (m >8). 7 Form= 8 it is necessary to adda 2  y 2  p 4 to the exponent ofPand form= 7 the terms a(a+ a 2 bp )  2 y  p 2 +a 3  3 y  p 3 to the exponent ofP, and the termab  2 y  p 2 to the exponent ofQ. The extra term 27ab 2 k  3 y  is to be added to the exponent ofPform= 7 andp= 3. 8 Form= 7, ap 2 − a 2 2 p 3 ≡ap 2 (modp 4 ), ap 3 ≡kp 3 (modp 4 ). Form= 7 andp= 3 the first of the above congruences has the extra terms 27(a 3 +abβk) on the left side. 9 Form= 8 it is necessary to add the terma  y 2  xp 4 to the exponent ofP, and form= 7 the termsx{a(a+ a 2 bp )  2 y  p 2 +a 3  y 3  p 3 }to the exponent ofP, with the extra term 27ab 2 k  3 y  x forp= 3, and the termab  y 2  xp 2 to the exponent ofQ. 8

From (12) we get 10 (13) [y, x] s =  ys+by  (x+b  x  p)  2 s  +x  s 3  p  p, 2 xs+ay  (x+b  x 2  p+b 2  x 3  p 2 )  2 s  + (bx 2 p+ 2b 2 x  x 2  p 2 )  3 s  +bx 2  s 4  p 2  p m−6  (m >8). 5.Transformation of the Groups.—The general groupGof Class I is spec- ified, in accordance with the relations (2) (11) by two integersa,bwhich (see (11)) are to be taken modp 3 , modp 2  Accordingly, respectively. setting a=a p λ b=b 1 p µ , 1 , where dv[a 1 , p] = 1, dv[b 1 , p (] = 1λ= 0,1,2,3;µ= 0,1,2), we have for the groupG=G(a, b) =G(a, b)(P, Q) the generational determi- nation: G(a, b) : ( Q −1 P Q=Q b 1 p µ+1 P 1+a 1 p m+λ−6 Q p 3 = 1, P p m−3 = 1. Not all of these groups however are distinct. Suppose that G(a, b)(P, Q)∼G(a 0 , b 0 )(P 0 , Q 0 ), by the correspondence , P C=  Q Q 0 1 , P 1 0  ,

where − Q 0 1 =Q 0y 0 P 0x 0 p m6 ,andP 1 0 =Q 0y P 0x , 10 Form= 8 it is necessary to add the term 21 axy  s 2  [ 1 3 y(2s−1)−1]p 4 to the exponent of P, and form= 7 the terms x n a 2  a+ a 2 b p  2s 3 −1 y−1  2 s  yp 2 + a 3! 3  s 2  y 2 −(2s−1)y+ 2  yp 3 y 2 + a 2 b 2 x  s 3  3s 2 −1 p 3 + a 2 2 b  s(s− 4! )1 2 (s−4) y−  3 s  yp 3 o with the extra terms 27abxy n b 3! k  2 s  y 2 −(2s−1)y+ 2  s 3  +x(b 2 k+a 2 )(2y 2 + 1)  s 3 o , forp= 3, to the exponent ofP, and the terms a 2 b  2s− 31 y−1  2 s  xyp 2 to the exponent ofQ. 9

withy 0 andxprime top. Since Q −1 P Q=Q bp P 1+ap m−6 , then Q 0 1 −1 P 1 0 Q 0 1 =Q 0 b 1 p P 0 1 +1ap m−6 , or in terms ofQ 0 , andP 0  y+b 0 xy 0 p+b 02  2 x  y 0 p 2 , x(1 +a 0 y 0 p m−6 ) +a 0 b 0  x 2  y 0 p m−5 +a 0 b 02  3 x  y 0 p m−4  = [y+by 0 p, x+ (ax+bx 0 p)p m−6 ] (m >8) and (14)by 0 ≡b 0 xy 0 +b 02  x 2  y 0 p(modp 2 ), (15)ax+bx 0 p≡a 0 y 0 x+a 0 b 0  x 2  y 0 p+a 0 b 02  3 x  y 0 p 2 (modp 3 ). The necessary and sufficient condition for the simple isomorphism of these two groupsG(a, b) andG(a 0 , b 0 the above congruences shall be consistent) is, that and admit of solution forx,y,x 0 andy 0  congruences may be written. The b 1 p µ ≡b 0 1 xp µ 0 +b 0 2 1  x 2  p 2µ 0 +1 (modp 2 ), a 1 xp λ +b 1 x 0 p µ+1 ≡ y 0 {a 0 1 xp λ 0 +a 0 1 b 0 1  x 2  p λ 0 +µ 0 +1 +a 0 1 b 0 1 2  3 x  p λ 0 +2µ 0 +2 }(modp 3 ). Sincedv[x, p] = 1 the first congruence givesµ=µ 0 andxmay always be so chosen thatb 1 = 1. We may choosey 0 in the second congruence so thatλ=λ 0 anda 1 = 1 except for the casesλ 0 ≥µ+ 1 =µ 0 + 1 when we will so choosex 0 thatλ= 3. The type groups of Class I form >8 11 are then given by (I)G(p λ , p µ ) :Q −1 P Q=Q p 1+µ P 1+p m−6+λ , Q p 3 = 1, P p m−3 = 1 µ;
µ µ 0 0 = = , , 1 1 , ,2; ;2 λ λ = 3 0 = ,1,2;λ≥ ! . Of the above groupsG(p λ , p µ ) the groups forµ= 2 have the cyclic sub- group{P}self-conjugate, while the groupG(p 3 , p 2 ) is the abelian group of type (m−3,3).

11 Form= 8 the additional termaypon the left side of the congruence (14) andappears G(1, p 2 ) andG(1, p The extra terms appearing in congruence) become simply isomorphic. (15) do not effect the result. Form= 7 the additional termayappears on the left side of (14) andG(1,1),G(1, p), andG(l, p 2 ) become simply isomorphic, alsoG(p, p) andG(p, p 2 ). 10

ClassII. 1.General relations. There is inGan operatorQ 1 such thatQ p 1 2 is contained in{P}whileQ 1 p is not. (1)Q p 1 2 =P hp 2 . The operatorsQ 1 andPeither generate a subgroupH 2 of orderp m−1 , or the entire groupG. Section1. 2.Groups with independent generators. Consider the first possibility in the above paragraph. There is inH 2 , a sub- groupH 1 of orderp m−2 , which contains{P}self-conjugately. 12 H 1 is generated byQ p 1 andP.H 2 containsH 1 self-conjugately and is itself self-conjugate inG. From these considerations 13 − (2)Q 1 −p P Q 1 p =P 1+kp m4 , (3)Q 1 −1 P Q=Q 1 βp P α 1 . 3.Determination ofH 1 andH 2 . From (2) we obtain (4) [yp, x] s =  syp, x  s+k  s 2  yp m−4  (m >4), and from (3) and (4) p [−p,1, p] =  α α 1 − −1 1 βp, α 1 p  1 + β 2 k α α 1 p 1 − − 1 1 p m−4  . 1 A comparison of the above equation with (2) shows that − 2 α α 1 1 p − 1 1 βp≡0 (modp), p − α p 1  1 + β 2a αk 1 1 p − − 1 1 p m−4  + α α 1 1 −1 1 βhp≡1 +kp m−4 (modp 3 ), m− and in turn α p 1 ≡1 (modp 2 ), α 1 ≡1 (modp) (m >5). Placingα 1 = 1 +α 2 pin the second congruence, we obtain as in Class I (5) (α 2 +βh)p 2 ≡kp m−4 (modp m−3 ) (m >5). 12 Burnside,Theory of Groups, Art. 54, p. 64. 13 Ibid., Art. 56, p. 66. 11