Groups of the Order p^m
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Groups of the Order p^m

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The Project Gutenberg EBook of Groups of Order p^m Which Contain Cyclic Subgroups of Order p^(m-3), by Lewis Irving Neikirk Copyright laws are changing all over the world. Be sure to check the copyright laws for your country before downloading or redistributing this or any other Project Gutenberg eBook. This header should be the first thing seen when viewing this Project Gutenberg file. Please do not remove it. Do not change or edit the header without written permission. Please read the "legal small print," and other information about the eBook and Project Gutenberg at the bottom of this file. Included is important information about your specific rights and restrictions in how the file may be used. You can also find out about how to make a donation to Project Gutenberg, and how to get involved.
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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 ORDERpmWHICH CONTAIN CYCLIC SUBGROUPS OF ORDERpm3
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 ORDERpm, WHICH CONTAIN CYCLIC SUBGROUPS OF ORDERp(m3)1 by lewis irving neikirk Introduction. The groups of orderpm, which contain self-conjugate cyclic subgroups of orderspm1, andpm2respectively, have been determined byBurnside,2and the number of groups of orderpm, which contain cyclic non-self-conjugate sub-groups of orderpm2has been given byMiller.3 Although in the present state of the theory, the actual tabulation of all groups of orderpmis impracticable, it is of importance to carry the tabulation as far as may be possible. In this paperall groups of orderpm(pbeing an odd prime)which contain cyclic subgroups of orderpm3and 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: (m3,3), (m3,2,1) and (m3,1,1,1), ofm. We denote byGan abstract groupGof orderpmcontaining operators of orderpm3no operator of order greater thanand pm3. LetPdenote one of these operators ofGof orderpm3. Thep3power of every operator inGis contained in the cyclic subgroup{P}, otherwiseGwould be of order greater than pmdivision into classes is effected by the following assumptions: complete . The I. There is inGat least one operatorQ1, such thatQp12is not contained in {P}. II. Thep2power of every operator inGis contained in{P}, and there is at least one operatorQ1, such thatQ1pis not contained in{P}. III. Thepth power of every operator inGis contained in{P}.
1Presented to the American Mathematical Society April 25, 1903. 2Theory of Groups of a Finite Order, pp. 75-81. 3Transactions, 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 II1II2II3II 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 operatorsPandQ1 brevity we set. For4 Qa1PbQc1Pd∙ ∙ ∙= [a, b, c, d,∙ ∙ ∙]. Then the operators ofGare given each uniquely in the form [y, x]xy0=0=,,11,,22,,p,p,3m311!. We have the relation (1)Qp13=Php3. There is inG, a subgroupH1of orderpm2, which contains{P}self-conjugate-ly.5The subgroupH1is generated byPand some operatorQy1PxofG; it then 2 containsQy1and is therefore generated byPandQ1p; it is also self-conjugate inH2={Qp1, P}of orderpm1, andH2is self-conjugate inG. From these considerations we have the equations6 (2)Q1p2P Q1p2=P1+kpm4, QpP Q1p=Qβp2 (3)1 1Pα1, (4)Q11P Q1=Qb1pPa1. 4With J. W.Young,On a certain group of isomorphisms, American Journal of Mathe-matics, vol. 25 (1903), p. 206. 5Burnside:Theory of Groups, Art. 54, p. 64. 6Ibid., Art. 56, p. 66. 5
2.Determination ofH1 of a formula for. Derivation[yp2, x]s.—From (2), by repeated multiplication we obtain [p2, x, p2] = [0, x(1 +kpm4)]; and by a continued use of this equation we have [yp2, x, yp2] = [0, x(1 +kpm4)y] = [0, x(1 +kypm4)] (m >4) and from this last equation, (5) [yp2, x]s=syp2, x{s+ksypm4}. 2 3.Determination ofH2 of a formula for. Derivation[yp, x]s.—It follows from (3) and (5) that [p2,1, p2] =β α1p111p2, αp11 +β2αkα1p111pm4(m >4). α Hence, by (2), β αp11p2m α110 ( odp3), αp11 +β2αkαp1111pm4+αβα1p111hp21 +kpm4(modpm). 3 From these congruences, we have form >6 α1p1 (modp3), α11 (modp2), and obtain, by setting 2 α1= 1 +α2p , the congruence (1 +αα2pp23)p(1α2+)p3kpm4(modpm3); 2 and so since
(α2+)p30 (modpm4),
+αp1 (1α22pp32)1 (modp2). 6
From the last congruences (6) (α2+)p3kpm4(modpm3). Equation (3) is now replaced by (7)Q1pP Q1p=Q1βp2P1+α2p2. From (7), (5), and (6) [yp, x, yp] =βxyp2, x{1 +α2yp2}+βk2xypm4. A continued use of this equation gives (8) [yp, x]s= [syp+β2sxyp2, xs+s2{α2xyp2+βkx2ypm4}+βk3sx2ypm4]. 4.Determination ofG.—From (4) and (8), [p,1, p] = [N p, ap1+M p2]. From the above equation and (7), ap11 (modp2), a11 (modp). Seta1= 1 +a2pand equation (4) becomes (9)Q11P Q1=Q1bpP1+a2p. From (9), (8) and (6) [p2,1, p2] ="(1 +aa22p)pp21bp,(1 +a2p)p2#, and from (1) and (2) (1 +aa22pp)p21bp0 (modp3), (1 +a2p)p2+bh(1 +aa22pp)p21p1 +kpm4(modpm3). By a reduction similar to that used before, (10) (a2+bh)p3kpm4(modpm3). 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 generatorQ1. From (9), (6), (8) and (10) [1, x]p3= [p3, xp3] = [0,(x+h)p3] (m >6), and, ifxbe so chosen that x+h0 (modpm6), Q1Pxis an operator of orderp3whosep2power is not contained in{P}. Let Q1Px=Q. The groupGis generated byQandP, where 3 Qp3= 1, Ppm= 1 . Placingh= 0 in (6) and (10) we find α2p3a2p3kpm4(modpm3). Letα2=αpm7, anda2=apm7 (7) and (9) are now replaced by. Equations (11)QpP Qp=Qβp2P1+αpm5, Q1P Q=QbpP1+apm6. As a direct result of the foregoing relations, the groups in this class corre-spond to the partition (m3, (11) we find3). From7 [y,1, y] = [byp,1 +aypm6] (m >8). It is important to notice that by placingy=pandp2in the preceding equation we find that8 bβ(modp), aαk(modp3) (m >7). A combination of the last equation with (8) yields9 (12) [y, x, y] = [bxyp+b2x2yp2, x(1 +aypm6) +abx2ypm5+ab2x3ypm4] (m >8). 7Form= 8 it is necessary to adda2y2p4to the exponent ofPand form= 7 the terms a(a+a2bp)2yp2+a33yp3to the exponent ofP, and the termab2yp2to the exponent ofQ. The extra term 27ab2k3yis to be added to the exponent ofPform= 7 andp= 3. 8Form= 7, ap2a22p3ap2(modp4), ap3kp3(modp4). Form= 7 andp= 3 the first of the above congruences has the extra terms 27(a3+abβk) on the left side. 9Form= 8 it is necessary to add the termay2xp4to the exponent ofP, and form= 7 the termsx{a(a+a2bp)2yp2+a3y3p3}to the exponent ofP, with the extra term 27ab2k3yx forp= 3, and the termaby2xp2to the exponent ofQ. 8
From (12) we get10 (13) [y, x]s=ys+by(x+bxp)2s+xs3pp, 2 xs+ay(x+bx2p+b2x3p2)2s+ (bx2p+ 2b2xx2p2)3s+bx2s4p2pm6(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 modp3, modp2 Accordingly, respectively. setting a=a pλb=b1pµ, 1, where dv[a1, p] = 1, dv[b1, 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) :(Q1P Q=Qb1pµ+1P1+a1pm+λ6 Qp3= 1, Ppm3= 1. Not all of these groups however are distinct. Suppose that G(a, b)(P, Q)G(a0, b0)(P0, Q0), by the correspondence , P C=QQ01, P10,
where Q01=Q0y0P0x0pm6,andP10=Q0yP0x, 10Form= 8 it is necessary to add the term21axys2[13y(2s1)1]p4to the exponent of P, and form= 7 the terms xna2a+a2bp2s31y12syp2+a3!3s2y2(2s1)y+ 2yp3 y2 +a2b2xs33s21p3+a22bs(s4!)12(s4)y3syp3o with the extra terms 27abxynb3!k2sy2(2s1)y+ 2s3+x(b2k+a2)(2y2+ 1)s3o, forp= 3, to the exponent ofP, and the termsa2b2s31y12sxyp2to the exponent ofQ. 9
withy0andxprime top. Since Q1P Q=QbpP1+apm6, then Q011P10Q01=Q0b1pP01+1apm6, or in terms ofQ0, andP0 y+b0xy0p+b022xy0p2, x(1 +a0y0pm6) +a0b0x2y0pm5 +a0b023xy0pm4= [y+by0p, x+ (ax+bx0p)pm6] (m >8) and (14)by0b0xy0+b02x2y0p(modp2), (15)ax+bx0pa0y0x+a0b0x2y0p+a0b023xy0p2(modp3). The necessary and sufficient condition for the simple isomorphism of these two groupsG(a, b) andG(a0, b0the above congruences shall be consistent) is, that and admit of solution forx,y,x0andy0 congruences may be written. The b1pµb01xpµ0+b021x2p2µ0+1(modp2), a1xpλ+b1x0pµ+1y0{a01xpλ0+a01b01x2pλ0+µ0+1+a01b0123xpλ0+2µ0+2}(modp3). Sincedv[x, p] = 1 the first congruence givesµ=µ0andxmay always be so chosen thatb1= 1. We may choosey0in the second congruence so thatλ=λ0anda1= 1 except for the casesλ0µ+ 1 =µ0+ 1 when we will so choosex0thatλ= 3. The type groups of Class I form >811are then given by (I)G(pλ, pµ) :Q1P Q=Qp1+µP1+pm6+λ, Qp3= 1, Ppm3= 1 µ; µµ00==,,11,,2;;2λλ=30=,1,2;λ!. Of the above groupsG(pλ, pµ) the groups forµ= 2 have the cyclic sub-group{P}self-conjugate, while the groupG(p3, p2) is the abelian group of type (m3,3).
11Form= 8 the additional termaypon the left side of the congruence (14) andappears G(1, p2) 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, p2) become simply isomorphic, alsoG(p, p) andG(p, p2). 10
ClassII. 1.General relations. There is inGan operatorQ1such thatQp12is contained in{P}whileQ1pis not. (1)Qp12=Php2. The operatorsQ1andPeither generate a subgroupH2of orderpm1, or the entire groupG. Section1. 2.Groups with independent generators. Consider the first possibility in the above paragraph. There is inH2, a sub-groupH1of orderpm2, which contains{P}self-conjugately.12H1is generated byQp1andP.H2containsH1self-conjugately and is itself self-conjugate inG. From these considerations13 (2)Q1pP Q1p=P1+kpm4, (3)Q11P Q=Q1βpPα1. 3.Determination ofH1andH2. From (2) we obtain (4) [yp, x]s=syp, xs+ks2ypm4(m >4), and from (3) and (4) p [p,1, p] =αα111βp, α1p1 +β2kαα1p111pm4. 1 A comparison of the above equation with (2) shows that 2 αα11p11βp0 (modp), p αp11 +β2aαk11p11pm4+αα1111βhp1 +kpm4(modp3), mand in turn αp11 (modp2), α11 (modp) (m >5). Placingα1= 1 +α2pin the second congruence, we obtain as in Class I (5) (α2+βh)p2kpm4(modpm3) (m >5). 12Burnside,Theory of Groups, Art. 54, p. 64. 13Ibid., Art. 56, p. 66. 11